KdV-type equations: Difference between revisions

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==Equations of Korteweg-de Vries type==
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<!--[if gte mso 9]><xml>
The ''equations of Korteweg-de Vries type'' are all nonlinear perturbations of the [[Airy equation]]. They take the general form
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<body lang=EN-US link=blue vlink=blue style='tab-interval:.5in'>
<center><math>\partial_t u + \partial_x^3 u + \partial_x P(u) = 0</math></center>


<div class=Section1>
where <math>u(t,x)</math> is a function of one space and one time variable, and <math>P(u)</math> is some polynomial of <math>u</math>. One can place various normalizing constants in front of the <math>u_{xxx}</math> and <math>P(u)</math> terms, but they can usually be scaled out. The function <math>u</math> and the polynomial <math>P</math> are usually assumed to be real. 


<h2 align=center style='text-align:center'>Equations of <span class=SpellE>Korteweg</span>
The <math>x</math> variable is usually assumed to live on the real line R (so there is some decay at infinity) or on the <span class="SpellE">torus</span> T (so the data is periodic). The half-line has also been studied, as has the case of periodic data with large period. It might be interesting to look at whether the periodicity assumption can be perturbed (e.g. quasi-periodic data); it is not clear whether the phenomena we see in the periodic problem are robust under perturbations, or are number-theoretic <span class="SpellE">artefacts</span> of perfect periodicity.
de <span class=SpellE>Vries</span> type</h2>


<div class=MsoNormal align=center style='text-align:center'>
== Specific equations ==


<hr size=2 width="100%" align=center>
Several special cases of KdV-type equations are of interest, including


</div>
* The [[Korteweg-de Vries equation|Korteweg-de Vries (KdV) equation]] ([[Korteweg-de Vries equation on R|on R]], [[Korteweg-de Vries equation on the half-line|on R^+]], or [[Korteweg-de Vries equation on T|on T]]), in which <math>P(u) = 3 u^2</math>.  This equation is [[completely integrable]].
* The [[modified Korteweg-de Vries equation|modified Korteweg-de Vries (mKdV) equation]] ([[Modified Korteweg-de Vries on R|on R]], [[generalized Korteweg-de Vries on the half-line|on R^+]], or [[Modified Korteweg-de Vries on T|on T]]), in which <math>P(u) = \pm 2 u^3</math>.  This equation is also [[completely integrable]].
* The [[generalized Korteweg-de Vries equation|generalized Korteweg-de Vries (gKdV) equation]], in which <math>P(u) = c u^{k+1}</math> for some constants c,k.  The cases k=1,2 are [[KdV]] and [[mKdV]] respectively.  The quartic [[gKdV-3 equation]] and the quintic (mass-critical) [[gKdV-4 equation]] are of special interest.  In general, these equations are not completely integrable.
* The linearized Korteweg-de Vries equation, in which <math>P(u) = c u</math> (i.e., the <math>k=0</math> case of the generalized Korteweg-de Vries (gKdV) equation).  This equations is linear in <math>u</math> and can be reduce to the simple form <math>u_T + u_{XXX} = 0</math> with the change of variables <math>X=x-t</math>, <math>T=t</math>.[http://mathoverflow.net/questions/23334/what-are-the-interesting-cases-of-the-generalized-korteweg-de-vries-equation]


<p class=MsoNormal align=center style='text-align:center'><a name=overview></a><b>Overview</b></p>
Drift terms <math>u_x</math> can be added, but they can be subsumed into the polynomial <math>P(u)</math> or eliminated by a [[Galilean]] transformation (except in the half-line case). Indeed, one can freely insert or remove any term of the form <math>a'(t) u_x</math> by shifting the <math>x</math> variable by <math>a(t)</math>, which is especially useful for periodic higher-order <span class="SpellE">gKdV</span> equations (setting <math>a'(t)</math> equal to the mean of <math>P(u(t))</math>).


<p>The <span class=SpellE>KdV</span> <span class=GramE>family of equations are</span>
The Korteweg-de Vries equation is also a member of the [[KdV hierarchy]].  One can also couple the KdV equation to other equations, creating for instance the [[nonlinear Schrodinger-Airy system]].
of the form </p>


<p align=center style='text-align:center'><span class=SpellE>u_t</span> + u<span
== History ==
class=GramE>_{</span>xxx} + P(u)_x = 0</p>


<p><span class=GramE>where</span> u(<span class=SpellE>x,t</span>) is a
Historically, these types of equations first arose in the study of 2D shallow wave propagation, but have since appeared as limiting cases of many dispersive models. Interestingly, the 2D shallow wave equation can also give rise to the <span class="SpellE">Boussinesq</span> or [[Cubic NLS on R|cubic NLS]] equation by making more limiting assumptions (in particular, weak nonlinearity and slowly varying amplitude).
function of one space and one time variable, and P(u) is some polynomial of
u.&nbsp; One can place various normalizing constants in front of the u<span
class=GramE>_{</span>xxx} and P(u) terms, but they can usually be scaled
out.&nbsp; The function u and the polynomial P are usually assumed to be real. </p>


<p>Historically, these types of equations first arose in the study of 2D
== Conservation laws, symmetries, and criticality ==
shallow wave propagation, but have since appeared as limiting cases of many
dispersive models.&nbsp; Interestingly, the 2D shallow wave equation can also
give rise to the <span class=SpellE>Boussinesq</span> or <a
href="schrodinger.html#Cubic NLS on R">1D NLS-3</a> equation by making more
limiting assumptions (in particular, weak nonlinearity and slowly varying
amplitude). </p>


<p>The x variable is usually assumed to live on the real line R (so there is
KdV-type equations on R or T always come with three conserved quantities:
some decay at infinity) or on the <span class=SpellE>torus</span> T (so the
data is periodic).&nbsp; The half-line has also been studied, as has the case
of periodic data with large period.&nbsp; It might be interesting to look at
whether the periodicity assumption can be perturbed (e.g. quasi-periodic data);
it is not clear whether the phenomena we see in the periodic problem are robust
under perturbations, or are number-theoretic <span class=SpellE>artefacts</span>
of perfect periodicity. </p>


<p>When <span class=GramE>P(</span>u) = c u^{k+1}, then the equation is
<center><math>Mass:=\int u dx, \| u \|_{L^2_x}^2 := \int u^2 dx, Hamiltonian:=\int  u_x^2 - V(u) dx</math></center>
referred to as generalized <span class=SpellE>gKdV</span> of order k, or <span
class=SpellE>gKdV</span>-k.&nbsp; <span class=GramE>gKdV-1</span> is the
original <span class=SpellE>Korteweg</span> de <span class=SpellE>Vries</span>


(<span class=SpellE>KdV</span>) equation, gKdV-2 is the modified <span
where <math>V</math> is a primitive of <math>P</math>. Note that the Hamiltonian is positive-definite in the [[defocussing]] cases (if u is real); thus the defocussing equations have a better chance of long-term existence. The mass has no definite sign and so is only useful in specific cases (e.g. perturbations of a <span class="SpellE">soliton</span>).
class=SpellE>KdV</span> (<span class=SpellE>mKdV</span>) equation.&nbsp; <span
class=SpellE>KdV</span> and <span class=SpellE>mKdV</span> are quite special,
being the only equations in this family which are completely <span
class=SpellE>integrable</span>. </p>


<p>If k is even, the sign of c is important.&nbsp; The c &lt; 0 case is known
In general, the above three quantities are the only conserved quantities available, but the [[KdV]] and [[mKdV]] equations come with infinitely many more such conserved quantities due to their [[completely integrable]] nature.
as the <span class=SpellE>defocussing</span> case, while c &gt; 0 is the <span
class=SpellE>focussing</span> case.&nbsp; When k is odd, the constant c can
always be scaled out, so we do not distinguish <span class=SpellE>focussing</span>


and <span class=SpellE>defocussing</span> in this case. </p>
The [[critical]] (or scaling) regularity is


<p>Drift terms <span class=SpellE>u_x</span> can be added, but they can be
<center><math>s_c = \frac{1}{2} - \frac{2}{k}.</math></center>
subsumed into the polynomial <span class=GramE>P(</span>u) or eliminated by a <span
class=SpellE>Gallilean</span> transformation [except in the half-line case].&nbsp;
Indeed, one can freely insert or remove any term of the form a'(t) <span
class=SpellE>u_x</span> by shifting the x variable by <span class=GramE>a(</span>t),
which is especially useful for periodic higher-order <span class=SpellE>gKdV</span>


equations (setting a'(t) equal to the mean of P(u(t))). </p>
In particular, [[KdV]], [[mKdV]], and [[GKdV-3 equation|gKdV-3]] are [[subcritical]] with respect to <math>L^2</math>, [[GKdV-4 equation|gKdV-4]] is <math>L^2</math> [[critical]], and all the other equations are <math>L^2</math> [[supercritical]]. Generally speaking, the potential energy term <math>V(u)</math> can be pretty much ignored in the sub-critical equations, needs to be dealt with carefully in the critical equation, and can completely dominate the Hamiltonian in the super-critical equations (to the point that blowup occurs if the equation is not defocussing</span>. Note that <math>H^1</math> is always a sub-critical regularity.


<p><span class=SpellE>KdV</span>-type equations on R or T always come with
The [[dispersion relation]] <math>\tau = \xi^3</math> is always increasing, which means that singularities always propagate to the left. In fact, high frequencies propagate leftward at extremely high speeds, which causes a smoothing effect if there is some decay in the initial data (<math>L^2</math> will do). On the other hand, KdV-type equations have the remarkable property of supporting localized <span class="SpellE">travelling</span> wave solutions known as [[soliton]]s, which propagate to the right. It is known that solutions to the [[completely integrable]] equations (i.e. [[KdV]] and [[mKdV]] always resolve to a superposition of solitons as <math>t \rightarrow \infty</math>, but it is an interesting open question as to whether the same phenomenon occurs for the other KdV-type equations.
three conserved quantities: </p>


<p align=center style='text-align:center'>Mass:&nbsp; \<span class=SpellE>int</span>
== Symplectic structure ==
u <span class=SpellE>dx</span> <br>
L^2: \<span class=SpellE>int</span> u^2 <span class=SpellE>dx</span> <br>


Hamiltonian: \<span class=SpellE>int</span> u_x^2 - <span class=GramE>V(</span>u)
A <span class="SpellE">KdV</span>-type equation can be viewed as a <span class="SpellE">symplectic</span> flow with the Hamiltonian defined above, and the <span class="SpellE">symplectic</span> form given by
<span class=SpellE>dx</span></p>


<p><span class=GramE>where</span> V is a primitive of P.&nbsp; Note that the
<center><math>(u,v) := \int u \partial_x^{-1} v dx</math>.</center>
Hamiltonian is positive-definite in the <span class=SpellE>defocussing</span>
cases (if u is real); thus the <span class=SpellE>defocussing</span> equations
have a better chance of long-term existence.&nbsp;&nbsp; The mass has no
definite sign and so is only useful in specific cases (e.g. perturbations of a <span
class=SpellE>soliton</span>). </p>


<p>In general, the above three quantities are the only conserved quantities
Thus <math>H^{-1/2}</math> is the natural Hilbert space in which to study the <span class="SpellE">symplectic</span> geometry of these flows. Unfortunately, the [[gKdV|gKdV-k]] equations are only locally well-posed in <math>H^{-1/2}</math> when <math>k=1</math>. In fact, the [[Korteweg-de_Vries_equation| standard KdV equation]] is bi-hamiltonian.
available, but the <a href="#kdv"><span class=SpellE>KdV</span></a> and <a
href="#mkdv"><span class=SpellE>mKdV</span></a> equations come with infinitely
many more such conserved quantities due to their completely <span class=SpellE>integrable</span>
nature. </p>


<p>The critical (or scaling) regularity is </p>


<p align=center style='text-align:center'><span class=SpellE>s_c</span> = 1/2 -
If <math>k</math> is even, the sign of <math>c</math> is important. The <math>c < 0</math> case is known as the <span class="SpellE">defocussing</span> case, while <math>c > 0</math> is the <span class="SpellE">focussing</span> case. When <math>k</math> is odd, the constant <math>c</math> can always be scaled out, so we do not distinguish <span class="SpellE">focussing</span> and <span class="SpellE">defocussing</span> in this case.
2/k.</p>


<p>In particular, <a href="#kdv"><span class=SpellE>KdV</span></a>, <a
==Estimates==
href="#mkdv"><span class=SpellE>mKdV</span></a>, and gKdV-3 are <span
class=SpellE>subcritical</span> with respect to L^2, gKdV-4 is L^2 critical,
and all the other equations are L^2 supercritical.&nbsp; Generally speaking,
the potential energy term V(u) can be pretty much ignored in the sub-critical
equations, needs to be dealt with carefully in the critical equation, and can
completely dominate the Hamiltonian in the super-critical equations (to the
point that blowup occurs if the equation is not <span class=SpellE>defocussing</span>).&nbsp;
Note that H^1 is always a sub-critical regularity. </p>


<p>The dispersion relation \<span class=SpellE>tau</span> = \xi^3 is always
The perturbation theory for the KdV-type equations rests on a number of [[Linear Airy estimates|linear]], [[Bilinear Airy estimates|bilinear]], [[Trilinear Airy estimates|trilinear]], or [[Multilinear Airy estimates|multilinear]] estimates for the [[Airy equation]].  These estimates involve a number of function space norms, such as the [[X^s,b spaces]]. See the page on [[Airy estimates]] for more details.
increasing, which means that singularities always propagate to the left.&nbsp;
In fact, high frequencies propagate leftward at extremely high speeds, which
causes a smoothing effect if there is some decay in the initial data (L^2 will
do).&nbsp; On the other hand, <span class=SpellE>KdV</span>-type equations have
the remarkable property of supporting localized <span class=SpellE>travelling</span>
wave solutions known as <span class=SpellE>solitons</span>, which propagate to
the right.&nbsp; It is known that solutions to the completely <span
class=SpellE>integrable</span> equations (i.e. <span class=SpellE>KdV</span>


and <span class=SpellE>mKdV</span>) always resolve to a superposition of <span
[[Category:Airy]]
class=SpellE>solitons</span> as t -&gt; infinity, but it is an interesting open
[[Category:Equations]]
question as to whether the same phenomenon occurs for the other <span
class=SpellE>KdV</span>-type equations. </p>
 
<p>A <span class=SpellE>KdV</span>-type equation can be viewed as a <span
class=SpellE>symplectic</span> flow with the Hamiltonian defined above, and the
 
<span class=SpellE>symplectic</span> form given by </p>
 
<p align=center style='text-align:center'>{<span class=GramE>u</span>, v} := \<span
class=SpellE>int</span> u <span class=SpellE>v_x</span> <span class=SpellE>dx</span>.</p>
 
<p>Thus H<span class=GramE>^{</span>-1/2} is the natural Hilbert space in which
to study the <span class=SpellE>symplectic</span> geometry of these
flows.&nbsp; Unfortunately, the <span class=SpellE>gKdV</span>-k equations are
only locally well-posed in H<span class=GramE>^{</span>-1/2} when k=1. </p>
 
<div class=MsoNormal align=center style='text-align:center'>
 
<hr size=2 width="100%" align=center>
 
</div>
 
<p class=MsoNormal align=center style='text-align:center'><a name=Airy></a><b>Airy
estimates</b></p>
 
<p>Solutions to the Airy equation and its perturbations are either estimated in
mixed space-time norms <span class=SpellE>L^q_t</span> <span class=SpellE>L^r_x</span>,
<span class=SpellE>L^r_x</span> <span class=SpellE>L^q_t</span>, or in X^{<span
class=SpellE>s<span class=GramE>,b</span></span>} spaces, defined by </p>
 
<p align=center style='text-align:center'><tt><span style='font-size:10.0pt'>||
u ||_{<span class=SpellE>s<span class=GramE>,b</span></span>} = ||
&lt;xi&gt;^s&nbsp; &lt;tau-xi^3&gt;^b \hat{u} ||_2.</span></tt></p>
 
<p>Linear space-time estimates in which the space norm is evaluated first are
known as <a href="#kdv_linear"><span class=SpellE>Strichartz</span> estimates</a>,
but these estimates only play a minor role in the theory.&nbsp; A more
important category of linear estimates are the smoothing estimates and maximal
function estimates.&nbsp;&nbsp;&nbsp; The X^{<span class=SpellE>s<span
class=GramE>,b</span></span>} spaces are used primarily for <a
href="#kdv_bilinear">bilinear estimates</a>, although more recently <a
href="#KdV_multilinear"><span class=SpellE>multilinear</span> estimates have
begun to appear</a>.&nbsp; These spaces and estimates first appear in the
context of the <span class=SpellE>Schrodinger</span> equation in [<a
href="references.html#Bo1993b">Bo1993b</a>], although the analogues spaces for
the wave equation appeared earlier [<a href="references.html#RaRe1982">RaRe1982</a>],
[<a href="references.html#Be1983">Be1983</a>] in the context of <span
class=SpellE>propogation</span> of singularities.&nbsp; See also [<a
href="references.html#Bo1993">Bo1993</a>], [<a href="references.html#KlMa1993">KlMa1993</a>].
 
</p>
 
<div class=MsoNormal align=center style='text-align:center'>
 
<hr size=2 width="100%" align=center>
 
</div>
 
<p class=MsoNormal align=center style='text-align:center'><a name="kdv_linear"></a><b>Linear
Airy estimates</b></p>
 
<ul type=disc>
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l4 level1 lfo2;tab-stops:list .5in'>If u is in X^{0,1/2+} on <b>R</b>,
    then</li>
 
<ul type=circle>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l4 level2 lfo2;tab-stops:list 1.0in'>u is in L^\<span
      class=SpellE>infty_t</span> L^2_x (energy estimate)</li>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l4 level2 lfo2;tab-stops:list 1.0in'><span class=SpellE>D_x</span>^{1/4}
      u is in L^4_t <span class=SpellE>BMO_x</span> (endpoint <span
      class=SpellE>Strichartz</span>) [<a href="references.html#KnPoVe1993">KnPoVe1993</a>]</li>
 
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l4 level2 lfo2;tab-stops:list 1.0in'><span class=SpellE>D_x</span>
      u is in L^\<span class=SpellE>infty_x</span> L^2_t (sharp Kato smoothing
      effect) [<a href="references.html#KnPoVe1993">KnPoVe1993</a>].&nbsp;
      Earlier versions of this estimate were obtained in [<a
      href="references.html#Ka1979b">Ka1979b</a>], [<a
      href="references.html#KrFa1983">KrFa1983</a>].</li>
 
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l4 level2 lfo2;tab-stops:list 1.0in'><span class=SpellE>D_x</span>^{-1/4}
      u is in L^4_x L^\<span class=SpellE>infty_t</span> (Maximal function) [<a
      href="references.html#KnPoVe1993">KnPoVe1993</a>], [<a
      href="references.html#KnRu1983">KnRu1983</a>]</li>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l4 level2 lfo2;tab-stops:list 1.0in'><span class=SpellE>D_x</span>^{-3/4-}
      u is in L^2_x L^\<span class=SpellE>infty_t</span> (L^2 maximal function)
      [<a href="references.html#KnPoVe1993">KnPoVe1993</a>]</li>
 
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l4 level2 lfo2;tab-stops:list 1.0in'><i>Remark</i>: Further
      estimates are available by <span class=SpellE>Sobolev</span>,
      differentiation, Holder, and interpolation.&nbsp; For instance:</li>
  <ul type=square>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l4 level3 lfo2;tab-stops:list 1.5in'><span class=SpellE>D_x</span>
      u is in L^2_{<span class=SpellE>x,t</span>} locally in space [<a
      href="references.html#Ka1979b">Ka1979b</a>] - use Kato and Holder (can
      also be proven directly by integration by parts)</li>
 
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l4 level3 lfo2;tab-stops:list 1.5in'>u is in L^2_{<span
      class=SpellE>x,t</span>} locally in time - use energy and Holder</li>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l4 level3 lfo2;tab-stops:list 1.5in'><span class=SpellE>D_x</span>^{3/4-}
      u is in L^8_x L^2_t locally in time - interpolate previous with Kato</li>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l4 level3 lfo2;tab-stops:list 1.5in'><span class=SpellE>D_x</span>^{1/6}
      u is in L^6_{<span class=SpellE>x,t</span>} - interpolate energy with
      endpoint <span class=SpellE>Strichartz</span> (or Kato with maximal)</li>
 
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l4 level3 lfo2;tab-stops:list 1.5in'><span class=SpellE>D_x</span><span
      class=GramE>^{</span>1/8} u is in L^8_t L^4_x - interpolate energy with
      endpoint <span class=SpellE>Strichartz</span>.&nbsp; (In particular, <span
      class=SpellE>D_x</span><span class=GramE>^{</span>1/8} u is also in
      L^4_{<span class=SpellE>x,t</span>}).</li>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l4 level3 lfo2;tab-stops:list 1.5in'>u is in L^8_{<span
      class=SpellE>x,t</span>}- use previous and <span class=SpellE>Sobolev</span>
 
      in space</li>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l4 level3 lfo2;tab-stops:list 1.5in'>If u is in
      X^{0,1/3+}, then u is in L^4_{<span class=SpellE>x,t</span>} [<a
      href="references.html#Bo1993b">Bo1993b</a>] - interpolate previous with
      the trivial identity X^{0,0} = L^2</li>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l4 level3 lfo2;tab-stops:list 1.5in'>If u is in
      X^{0,1/4+}, then <span class=SpellE>D_x</span>^{1/2} u is in L^4_x L^2_t
      [<a href="references.html#Bo1993b">Bo1993b</a>] - interpolate Kato with
      X^{0,0} = L^2</li>
 
  </ul>
</ul>
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l4 level1 lfo2;tab-stops:list .5in'>If u is in X^{0,1/2+} on <b>T</b>,
    then</li>
<ul type=circle>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l4 level2 lfo2;tab-stops:list 1.0in'><span class=GramE>u</span>
      is in L^\<span class=SpellE>infty_t</span> L^2_x (energy estimate).&nbsp;
 
      This is also true in the large period case.</li>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l4 level2 lfo2;tab-stops:list 1.0in'><span class=GramE>u</span>
      is in L^4_{<span class=SpellE>x,t</span>} locally in time (in fact one
      only needs u in X^{0,1/3} for this) [<a href="references.html#Bo1993b">Bo1993b</a>].</li>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l4 level2 lfo2;tab-stops:list 1.0in'><span class=SpellE>D_x</span><span
      class=GramE>^{</span>-\<span class=SpellE>eps</span>} u is in L^6_{<span
      class=SpellE>x,t</span>} locally in time. [<a
      href="references.html#Bo1993b">Bo1993b</a>].&nbsp; It is conjectured that
      this can be improved to L^8_{<span class=SpellE>x<span class=GramE>,t</span></span>}.</li>
 
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l4 level2 lfo2;tab-stops:list 1.0in'><i>Remark</i>: there
      is no smoothing on the circle, so one can never gain regularity.</li>
</ul>
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l4 level1 lfo2;tab-stops:list .5in'>If u is in X^{0,1/2} on a
    circle with large period \lambda, then</li>
<ul type=circle>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l4 level2 lfo2;tab-stops:list 1.0in'><span class=GramE>u</span>
      is in L^4_{<span class=SpellE>x,t</span>} locally in time, with a bound
      of \lambda^{0+}.</li>
 
  <ul type=square>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l4 level3 lfo2;tab-stops:list 1.5in'>In fact, when u has frequency
      N, the constant is like \lambda^{0+} (N<span class=GramE>^{</span>-1/8}
      + \lambda^{-1/4}), which recovers the 1/8 smoothing effect on the line
      in L^4 mentioned earlier. [<a href="references.html#CoKeStTaTk-p2">CoKeStTkTa-p2</a>]</li>
  </ul>
</ul>
</ul>
 
<h4 align=center style='text-align:center'>
 
<hr size=2 width="100%" align=center>
 
</h4>
 
<p class=MsoNormal align=center style='text-align:center'><a name="kdv_bilinear"></a><b>Bilinear
Airy estimates</b></p>
 
<ul type=disc>
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l23 level1 lfo3;tab-stops:list .5in'>The key algebraic fact is</li>
</ul>
 
<p class=MsoNormal align=center style='text-align:center'><tt><span
style='font-size:10.0pt'>\xi_1^3 + \xi_2^3 + \xi_3^3 = 3 \xi_1 \xi_2 \xi_3</span></tt>
<br>
<tt><span style='font-size:10.0pt'>(whenever \xi_1 + \xi_2 + \xi_3 = 0)</span></tt></p>
 
<ul type=disc>
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l11 level1 lfo4;tab-stops:list .5in'>The -3/4+ estimate [<a
    href="references.html#KnPoVe1996">KnPoVe1996</a>] on <b>R</b>:</li>
</ul>
 
<p class=MsoNormal align=center style='text-align:center'><tt><span
style='font-size:10.0pt'>|| (<span class=SpellE>uv</span><span class=GramE>)_</span>x
||_{-3/4+, -1/2+} &lt;~ || u ||_{-3/4+, 1/2+}&nbsp; || v ||_{-3/4+, 1/2+}</span></tt></p>
 
<ul type=disc>
<ul type=circle>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l10 level2 lfo5;tab-stops:list 1.0in'>The above estimate
      fails at the endpoint -3/4.&nbsp; [<a href="references.html#NaTkTs-p">NaTkTs2001</a>]</li>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l10 level2 lfo5;tab-stops:list 1.0in'>As a corollary of
      this estimate we have the -3/8+ estimate [<a
      href="references.html#CoStTk1999">CoStTk1999</a>] on <b>R</b>: If u and v
      have no low frequencies ( |\xi| &lt;~ 1 ) then</li>
 
</ul>
</ul>
 
<p class=MsoNormal align=center style='text-align:center'><tt><span
style='font-size:10.0pt'>|| (<span class=SpellE>uv</span><span class=GramE>)_</span>x
||_{0, -1/2+} &lt;~ || u ||_{-3/8+, 1/2+}&nbsp; || v ||_{-3/8+, 1/2+}</span></tt></p>
 
<ul type=disc>
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l27 level1 lfo6;tab-stops:list .5in'>The -1/2 estimate [<a
    href="references.html#KnPoVe1996">KnPoVe1996</a>] on <b>T</b>: if <span
    class=SpellE>u,v</span> have mean zero, then for all s &gt;= -1/2</li>
 
</ul>
 
<p class=MsoNormal align=center style='text-align:center'><tt><span
style='font-size:10.0pt'>|| (<span class=SpellE>uv</span><span class=GramE>)_</span>x
||_{s, -1/2} &lt;~ || u ||_{s, 1/2}&nbsp; || v ||_{s, 1/2}</span></tt></p>
 
<ul type=disc>
<ul type=circle>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l12 level2 lfo7;tab-stops:list 1.0in'>The above estimate
      fails for s &lt; -1/2.&nbsp; Also, one cannot replace 1/2, -1/2 by 1/2+,
      -1/2+.&nbsp; [<a href="references.html#KnPoVe1996">KnPoVe1996</a>]</li>
 
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l12 level2 lfo7;tab-stops:list 1.0in'>This estimate also
      holds in the large period case if one is willing to lose a power of
      \lambda^{0+} in the constant. [<a href="references.html#CoKeStTaTk-p2">CoKeStTkTa-p2</a>]</li>
</ul>
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l12 level1 lfo7;tab-stops:list .5in'><i>Remark</i>: In principle,
    a complete list of bilinear estimates could be obtained from [<a
    href="references.html#Ta-p2">Ta-p2</a>].</li>
</ul>
 
<div class=MsoNormal align=center style='text-align:center'>
 
<hr size=2 width="100%" align=center>
 
</div>
 
<p class=MsoNormal align=center style='text-align:center'><a
name="kdv_trilinear"></a><span class=SpellE><b>Trilinear</b></span><b> Airy
estimates</b></p>
 
<ul type=disc>
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l3 level1 lfo8;tab-stops:list .5in'>The key algebraic fact is
    (various permutations of)</li>
</ul>
 
<p class=MsoNormal align=center style='text-align:center'><tt><span
style='font-size:10.0pt'>\xi_1^3 + \xi_2^3 + \xi_3^3 + \xi_4^3 = 3
(\xi_1+\xi_4) (\xi_2+\xi_4) (\xi_3+\xi_4)</span></tt> <br>
 
<tt><span style='font-size:10.0pt'>(whenever \xi_1 + \xi_2 + \xi_3 + \xi_4 = 0)</span></tt></p>
 
<ul type=disc>
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l16 level1 lfo9;tab-stops:list .5in'>The 1/4 estimate [<a
    href="references.html#Ta-p2">Ta-p2</a>] on <b>R</b>:</li>
</ul>
 
<p class=MsoNormal align=center style='text-align:center'><tt><span
style='font-size:10.0pt'>|| (<span class=SpellE>uvw</span><span class=GramE>)_</span>x
||_{1/4, -1/2+} &lt;~ || u ||_{1/4, 1/2+}&nbsp; || v ||_{1/4, 1/2+} || w
||_{1/4, 1/2+}</span></tt></p>
 
<p class=MsoNormal style='mso-margin-top-alt:auto;margin-right:.5in;mso-margin-bottom-alt:
auto;margin-left:.5in'>The 1/4 is sharp [<a href="references.html#KnPoVe1996">KnPoVe1996</a>].<span
style='mso-spacerun:yes'>  </span>We also have</p>
 
<p class=MsoNormal align=center style='text-align:center'><tt><span
style='font-size:10.0pt'>|| <span class=SpellE><span class=GramE>uv<u>w</u></span></span>
||_{-1/4, -5/12+} &lt;~ || u ||_{-1/4, 7/12+}&nbsp; || v ||_{-1/4, 7/12+} || w
||_{-1/4, 7/12+}</span></tt></p>
 
<p class=MsoNormal style='mso-margin-top-alt:auto;margin-right:.5in;mso-margin-bottom-alt:
auto;margin-left:.5in'><span class=GramE>see</span> [<span class=SpellE>Cv</span>-p].</p>
 
<ul type=disc>
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l15 level1 lfo10;tab-stops:list .5in'>The 1/2 estimate [<a
    href="references.html#CoKeStTaTk-p3">CoKeStTkTa-p3</a>] on <b>T</b>: if <span
    class=SpellE>u,v,w</span> have mean zero, then</li>
 
</ul>
 
<p class=MsoNormal align=center style='text-align:center'><tt><span
style='font-size:10.0pt'>|| (<span class=SpellE>uvw</span><span class=GramE>)_</span>x
||_{1/2, -1/2} &lt;~ || u ||_{1/2, 1/2*}&nbsp; || v ||_{1/2, 1/2*} || w
||_{1/2, 1/2*}</span></tt></p>
 
<p class=MsoNormal style='mso-margin-top-alt:auto;margin-right:.5in;mso-margin-bottom-alt:
auto;margin-left:.5in'>The 1/2 is sharp [<a href="references.html#KnPoVe1996">KnPoVe1996</a>].</p>
 
<ul type=disc>
 
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l28 level1 lfo11;tab-stops:list .5in'><i>Remark</i>: the <span
    class=SpellE>trilinear</span> estimate always needs one more derivative of
    regularity than the bilinear estimate; this is consistent with the heuristics
    from the Miura transform from <span class=SpellE>mKdV</span> to <span
    class=SpellE>KdV</span>.</li>
</ul>
 
<div class=MsoNormal align=center style='text-align:center'>
 
<hr size=2 width="100%" align=center>
 
</div>
 
<p class=MsoNormal align=center style='text-align:center'><a
name="KdV_multilinear"></a><span class=SpellE><b>Multilinear</b></span><b> Airy
estimates</b></p>
 
<ul type=disc>
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l13 level1 lfo12;tab-stops:list .5in'>We have the <span
    class=SpellE>quintilinear</span> estimate on <b>R</b>: [<a
    href="references.html#CoKeStTaTk-p2">CoKeStTkTa-p2</a>]</li>
 
</ul>
 
<p class=MsoNormal align=center style='text-align:center'><tt><span
style='font-size:10.0pt'>\<span class=SpellE>int</span> u^3 v^2 <span
class=SpellE>dx</span> <span class=SpellE>dt</span> &lt;~ || u ||<span
class=GramE>_{</span>1/4+,1/2+}^3 || v ||_{-3/4+,1/2+}^2</span></tt></p>
 
<ul type=disc>
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l5 level1 lfo13;tab-stops:list .5in'>The analogue for this on <b>T</b>
 
    is: [<a href="references.html#CoKeStTaTk-p2">CoKeStTkTa-p2</a>, <a
    href="references.html#CoKeStTaTk-p3">CoKeStTkTa-p3</a>]</li>
</ul>
 
<p class=MsoNormal align=center style='text-align:center'><tt><span
style='font-size:10.0pt'>\<span class=SpellE>int</span> u^3 v^2 <span
class=SpellE>dx</span> <span class=SpellE>dt</span> &lt;~ || u ||<span
class=GramE>_{</span>1/2,1/2*}^3 || v ||_{-1/2,1/2*}^2</span></tt></p>
 
<p class=MsoNormal style='mso-margin-top-alt:auto;margin-right:.5in;mso-margin-bottom-alt:
auto;margin-left:.5in'>In fact, this estimate also holds for large period, but
a loss of lambda<span class=GramE>^{</span>0+}.</p>
 
<div class=MsoNormal align=center style='text-align:center'>
 
<hr size=2 width="100%" align=center>
 
</div>
 
<p class=MsoNormal align=center style='text-align:center'><a name=kdv></a><b>The
<span class=SpellE>KdV</span> equation</b></p>
 
<p>The <span class=SpellE>KdV</span> equation is </p>
 
<p align=center style='text-align:center'><span class=SpellE>u_t</span> + <span
class=SpellE>u_xxx</span> + u <span class=SpellE>u_x</span> = 0.</p>
 
<p>It is completely <span class=SpellE>integrable</span>, and has infinitely
many conserved quantities.&nbsp; Indeed, for each non-negative integer k, there
is a conserved quantity which is roughly equivalent to the <span class=SpellE>H^k</span>
norm of u. </p>
 
<p>The <span class=SpellE>KdV</span> equation has been studied on the <a
href="#kdv_on_R">line</a>, the <a href="#kdv_on_T">circle</a>, and the <a
href="#KdV_on_R+">half-line</a>. </p>
 
<div class=MsoNormal align=center style='text-align:center'>
 
<hr size=2 width="100%" align=center>
 
</div>
 
<p class=MsoNormal align=center style='text-align:center'><a name="kdv_on_R"></a><span
class=SpellE><b>KdV</b></span><b> on R</b></p>
 
<ul type=disc>
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l17 level1 lfo14;tab-stops:list .5in'>Scaling is <span
    class=SpellE>s_c</span> = -3/2.</li>
 
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l17 level1 lfo14;tab-stops:list .5in'>LWP in <span class=SpellE>H^s</span>
    for s &gt;= -3/4 [<span class=SpellE>CtCoTa</span>-p], using a modified
    Miura transform and the <a href="#mKdV_on_R"><span class=SpellE>mKdV</span>
    theory</a>.&nbsp; This is despite the failure of the key bilinear estimate
    [<a href="references.html#NaTkTs-p">NaTkTs2001</a>]</li>
 
<ul type=circle>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l17 level2 lfo14;tab-stops:list 1.0in'>For s within a
      logarithm for s=-3/4 [<span class=SpellE>MurTao</span>-p].</li>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l17 level2 lfo14;tab-stops:list 1.0in'>Was proven for s
      &gt; -3/4 [<a href="references.html#KnPoVe1996">KnPoVe1996</a>].</li>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l17 level2 lfo14;tab-stops:list 1.0in'>Was proven for s
      &gt; -5/8 in [<a href="references.html#KnPoVe1993b">KnPoVe1993b</a>].</li>
 
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l17 level2 lfo14;tab-stops:list 1.0in'>Was proven for s
      &gt;= 0 in [<a href="references.html#Bo1993b">Bo1993b</a>].</li>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l17 level2 lfo14;tab-stops:list 1.0in'>Was proven for s
      &gt; 3/4 in [<a href="references.html#KnPoVe1993">KnPoVe1993</a>].</li>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l17 level2 lfo14;tab-stops:list 1.0in'>Was proven for s
      &gt; 3/2 in [<a href="references.html#BnSm1975">BnSm1975</a>], [<a
      href="references.html#Ka1975">Ka1975</a>], [<a
      href="references.html#Ka1979">Ka1979</a>], [<a
      href="references.html#GiTs1989">GiTs1989</a>], [<a
      href="references.html#Bu1980">Bu1980</a>]<span class=GramE>,&nbsp; ....</span></li>
 
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l17 level2 lfo14;tab-stops:list 1.0in'>One has local ill-<span
      class=SpellE><span class=GramE>posedness</span></span><span class=GramE>(</span>in
      the sense that the map is not uniformly continuous) for s &lt; -3/4 (in
      the complex setting) by <span class=SpellE>soliton</span> examples [<a
      href="references.html#KnPoVe-p"><span class=SpellE>KnPoVe</span>-p</a>].</li>
  <ul type=square>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l17 level3 lfo14;tab-stops:list 1.5in'>For real <span
      class=SpellE>KdV</span> this has also been established in [<span
      class=SpellE>CtCoTa</span>-p], by the Miura transform and the <a
      href="#mKdV_on_R">corresponding result for <span class=SpellE>mKdV</span></a>.</li>
 
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l17 level3 lfo14;tab-stops:list 1.5in'>Below -3/4 the
      solution map was known to not be C^3 [<a href="references.html#Bo1993b">Bo1993b</a>],
      [<a href="references.html#Bo1997">Bo1997</a>]; this was refined to C^2
      in [<a href="references.html#Tz1999b">Tz1999b</a>].</li>
  </ul>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l17 level2 lfo14;tab-stops:list 1.0in'>When the initial
      data is a real, rapidly decreasing measure one has a global smooth
      solution for t &gt; 0 [<a href="referencs.html#Kp1993">Kp1993</a>].&nbsp;
 
      Without the rapidly decreasing hypothesis one can still construct a
      global weak solution [<a href="references.html#Ts1989">Ts1989</a>]</li>
</ul>
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l17 level1 lfo14;tab-stops:list .5in'>GWP in <span class=SpellE>H^s</span>
    for s &gt; -3/4 (if u is real) [<a href="references.html#CoKeStTaTk2003">CoKeStTkTa2003</a>].</li>
 
<ul type=circle>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l17 level2 lfo14;tab-stops:list 1.0in'>Was proven for s
      &gt; -3/10 in [<a href="references.html#CoKeStTkTa2001">CoKeStTkTa2001</a>]</li>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l17 level2 lfo14;tab-stops:list 1.0in'>Was proven for
      s&gt;= 0 in [<a href="references.html#Bo1993b">Bo1993b</a>].&nbsp; Global
      weak solutions in L^2 were constructed in [<a
      href="references.html#Ka1983">Ka1983</a>], [<a
      href="references.html#KrFa1983">KrFa1983</a>], and were shown to obey the
      expected local smoothing estimate.&nbsp; These weak solutions were shown
      to be unique in [<a href="references.html#Zh1997b">Zh1997b</a>]</li>
 
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l17 level2 lfo14;tab-stops:list 1.0in'>Was proven for
      s&gt;= 1 in [<a href="references.html#KnPoVe1993">KnPoVe1993</a>].</li>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l17 level2 lfo14;tab-stops:list 1.0in'>Was proven for
      s&gt;= 2 in [<a href="references.html#BnSm1975">BnSm1975</a>], [<a
      href="references.html#Ka1975">Ka1975</a>], [<a
      href="references.html#Ka1979">Ka1979</a>]<span class=GramE>, ....</span></li>
 
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l17 level2 lfo14;tab-stops:list 1.0in'><i>Remark</i>: In
      the complex setting GWP fails for large data with Fourier support on the
      half-line [Bona/<span class=SpellE>Winther</span>?], [<span class=SpellE>Birnir</span>]<span
      class=GramE>, ????</span>.&nbsp; This result extends to a wide class of
      dispersive PDE.</li>
</ul>
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l17 level1 lfo14;tab-stops:list .5in'>By use of the inverse
    scattering transform one can show that smooth solutions eventually resolve
    into <span class=SpellE>solitons</span>, that two colliding <span
    class=SpellE>solitons</span> emerge as (slightly phase shifted) <span
    class=SpellE>solitons</span>, etc.</li>
 
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l17 level1 lfo14;tab-stops:list .5in'><span class=SpellE>Solitons</span>
    are <span class=SpellE>orbitally</span> H^1 stable [<a
    href="references.html#Bj1972">Bj1972</a>]</li>
<ul type=circle>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l17 level2 lfo14;tab-stops:list 1.0in'>In <span
      class=SpellE>H^s</span>, 0 &lt;= s &lt; 1, the orbital stability of <span
      class=SpellE>solitons</span> is at most polynomial (the distance to the
      ground state manifold in <span class=SpellE>H^s</span> norm grows like at
      most O(t^{1-s+}) in time) [<span class=SpellE>RaySt</span>-p]</li>
 
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l17 level2 lfo14;tab-stops:list 1.0in'>In L^2, orbital
      stability has been obtained in [<a href="references.html#MeVe2003">MeVe2003</a>].</li>
</ul>
</ul>
 
<p class=MsoNormal>The <span class=SpellE>KdV</span> equation can also be
generalized to a 2x2 system </p>
 
<p class=MsoNormal align=center style='text-align:center'><span class=SpellE><tt><span
style='font-size:10.0pt'>u_t</span></tt></span><tt><span style='font-size:10.0pt'>
 
+ <span class=SpellE>u_xxx</span> + a_3 <span class=SpellE>v_xxx</span> + u <span
class=SpellE>u_x</span> + a_1 v <span class=SpellE>v_x</span> + a_2 (<span
class=SpellE>uv</span><span class=GramE>)_</span>x = 0</span></tt> <br>
 
<tt><span style='font-size:10.0pt'>b_1 <span class=SpellE>v_t</span> + <span
class=SpellE>v_xxx</span> + b_2 a_3 <span class=SpellE>u_xxx</span> + v <span
class=SpellE>v_x</span> + b_2 a_2 u <span class=SpellE>u_x</span> + b_2 a_1 (<span
class=SpellE>uv</span>)_x + r <span class=SpellE>v_x</span></span></tt></p>
 
<p class=MsoNormal><span class=GramE>where</span> b_1,b_2 are positive
constants and a_1,a_2,a_3,r are real constants.&nbsp; This system was
introduced in [<a href="references.html#GeaGr1984">GeaGr1984</a>] to study
strongly interacting pairs of weakly nonlinear long waves, and studied further
in [<a href="references.html#BnPoSauTm1992">BnPoSauTm1992</a>].&nbsp; In [<a
href="references.html#AsCoeWgg1996">AsCoeWgg1996</a>] it was shown that this
system was also globally well-posed on L^2. <br>
It is an interesting question as to whether these results can be pushed further
to match the <span class=SpellE>KdV</span> theory; the apparent lack of
complete <span class=SpellE>integrability</span> in this system (for generic
choices of parameters <span class=SpellE>b_i</span>, <span class=SpellE>a_i</span>,
 
<span class=GramE>r</span>) suggests a possible difficulty. </p>
 
<div class=MsoNormal align=center style='text-align:center'>
 
<hr size=2 width="100%" align=center>
 
</div>
 
<p class=MsoNormal align=center style='text-align:center'><a name="KdV_on_R+"></a><span
class=SpellE><b>KdV</b></span><b> on R^+</b></p>
 
<ul type=disc>
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l29 level1 lfo15;tab-stops:list .5in'>The <span class=SpellE>KdV</span>
 
    Cauchy-boundary problem on the half-line is</li>
</ul>
 
<p class=MsoNormal align=center style='text-align:center'><span class=SpellE>u_t</span>
+ u<span class=GramE>_{</span>xxx} + <span class=SpellE>u_x</span> + u <span
class=SpellE>u_x</span> = 0;&nbsp; u(x,0) = u_0(x);&nbsp; u(0,t) = h(t)</p>
 
<p class=MsoNormal style='margin-left:.5in'>The sign of u<span class=GramE>_{</span>xxx}
is important (it makes the influence of the boundary x=0 mostly negligible),
the sign of u <span class=SpellE>u_x</span> is not.&nbsp; The drift term <span
class=SpellE>u_x</span> appears naturally from the derivation of <span
class=SpellE>KdV</span> from fluid mechanics.&nbsp; (On R, this drift term can
be eliminated by a <span class=SpellE>Gallilean</span> transform, but this is
not available on the half-line). </p>
 
<ul type=disc>
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l8 level1 lfo16;tab-stops:list .5in'>Because one is restricted to
    the half-line, it becomes a little tricky to use the Fourier
    transform.&nbsp; One approach is to use the Fourier-<span class=SpellE>Laplace</span>
    transform instead.</li>
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l8 level1 lfo16;tab-stops:list .5in'>Some compatibility conditions
    between u_0 and h are needed.&nbsp; The higher the regularity, the more
    compatibility conditions are needed.&nbsp; If the initial data u_0 is in <span
    class=SpellE>H^s</span>, then by scaling heuristics the natural space for
    h is in H<span class=GramE>^{</span>(s+1)/3}.&nbsp; (Remember that time
    has dimensions <i>length</i>^3).</li>
 
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l8 level1 lfo16;tab-stops:list .5in'>LWP is known for initial
    data in <span class=SpellE>H^s</span> and boundary data in H<span
    class=GramE>^{</span>(s+1)/3} for s &gt;= 0 [<span class=SpellE>CoKe</span>-p],
    assuming compatibility.&nbsp; The drift term may be omitted because of the
    time localization.</li>
<ul type=circle>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l8 level2 lfo16;tab-stops:list 1.0in'>For s &gt; 3/4 this
      was proven in [<a href="references.html#BnSuZh-p"><span class=SpellE>BnSuZh</span>-p</a>]
      (assuming that there is a drift term).</li>
 
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l8 level2 lfo16;tab-stops:list 1.0in'>Was proven for data
      in sufficiently weighted H^1 spaces in [<a href="references.html#Fa1983">Fa1983</a>].</li>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l8 level2 lfo16;tab-stops:list 1.0in'>From the real line
      theory one might expect to lower this to -3/4, but there appear to be
      technical difficulties with this.</li>
</ul>
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l8 level1 lfo16;tab-stops:list .5in'>GWP is known for initial
    data in L^2 and boundary data in H<span class=GramE>^{</span>7/12},
    assuming compatibility.</li>
<ul type=circle>
 
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l8 level2 lfo16;tab-stops:list 1.0in'>for initial data in
      H^1 and boundary data in H^{5/6}_loc this was proven in [<a
      href="references.html#BnSuZh-p"><span class=SpellE>BnSuZh</span>-p</a>]</li>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l8 level2 lfo16;tab-stops:list 1.0in'>Was proven for smooth
      data in [<a href="references.html#BnWi1983">BnWi1983</a>]</li>
</ul>
</ul>
 
<div class=MsoNormal align=center style='text-align:center'>
 
<hr size=2 width="100%" align=center>
 
</div>
 
<p class=MsoNormal align=center style='text-align:center'><a name="kdv_on_T"></a><span
class=SpellE><b>KdV</b></span><b> on T</b></p>
 
<ul type=disc>
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l6 level1 lfo17;tab-stops:list .5in'>Scaling is <span
    class=SpellE>s_c</span> = -3/2.</li>
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l6 level1 lfo17;tab-stops:list .5in'>C^0 LWP in <span
    class=SpellE>H^s</span> for s &gt;= -1, assuming u is real [<span
    class=SpellE>KpTp</span>-p]</li>
 
<ul type=circle>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l6 level2 lfo17;tab-stops:list 1.0in'>C^0 LWP in <span
      class=SpellE>H^s</span> for s &gt;= -5/8 follows (at least in principle)
      from work on the <span class=SpellE>mKdV</span> equation by [Takaoka and <span
      class=SpellE>Tsutsumi</span>?]</li>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l6 level2 lfo17;tab-stops:list 1.0in'>Analytic LWP in <span
      class=SpellE>H^s</span> for s &gt;= -1/2, in the complex case [<a
      href="references.html#KnPoVe1996">KnPoVe1996</a>].&nbsp; In addition to
      the usual bilinear estimate, one needs a linear estimate to keep the
      solution in <span class=SpellE>H^s</span> for t&gt;0.</li>
 
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l6 level2 lfo17;tab-stops:list 1.0in'>Analytic LWP was
      proven for s &gt;= 0 in [<a href="references.html#Bo1993b">Bo1993b</a>].</li>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l6 level2 lfo17;tab-stops:list 1.0in'>Analytic ill <span
      class=SpellE>posedness</span> at s&lt;-1/2, even in the real case [<a
      href="references.html#Bo1997">Bo1997</a>]</li>
  <ul type=square>
 
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l6 level3 lfo17;tab-stops:list 1.5in'>This has been
      refined to failure of uniform continuity at s&lt;-1/2 [<span
      class=SpellE>CtCoTa</span>-p]</li>
  </ul>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l6 level2 lfo17;tab-stops:list 1.0in'>Remark: s=-1/2 is the
      <span class=SpellE>symplectic</span> regularity, and so the machinery of
      infinite-dimensional <span class=SpellE>symplectic</span> geometry
      applies once one has a continuous flow, although there are some
      technicalities involving approximating <span class=SpellE>KdV</span> by a
      suitable <span class=SpellE>symplectic</span> finite-dimensional
      flow.&nbsp; In particular one has <span class=SpellE>symplectic</span>
 
      non-squeezing [CoKeStTkTa-p9], [<a href="references.html#Bo1999">Bo1999</a>].</li>
</ul>
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l6 level1 lfo17;tab-stops:list .5in'>C^0 GWP in <span
    class=SpellE>H^s</span> for s &gt;= -1, in the real case [<span
    class=SpellE>KpTp</span>-p].</li>
<ul type=circle>
 
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l6 level2 lfo17;tab-stops:list 1.0in'>Analytic GWP in <span
      class=SpellE>H^s</span> in the real case for s &gt;= -1/2 [<a
      href="references.html#CoKeStTaTk-p2">CoKeStTkTa-p2</a>]; see also [<a
      href="references.html#CoKeStTaTk-p3">CoKeStTkTa-p3</a>].</li>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l6 level2 lfo17;tab-stops:list 1.0in'>A short proof for the
      s &gt; -3/10 case is in [<a href="references.html#CoKeStTaTk-p2a">CoKeStTkTa-p2a</a>]</li>
 
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l6 level2 lfo17;tab-stops:list 1.0in'>Was proven for s
      &gt;= 0 in [<a href="references.html#Bo1993b">Bo1993b</a>].</li>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l6 level2 lfo17;tab-stops:list 1.0in'>GWP for real initial
      data which are measures of small norm [<a href="references.html#Bo1997">Bo1997</a>]
      <span class=GramE>The</span> small norm restriction is presumably
      technical.</li>
  <ul type=square>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l6 level3 lfo17;tab-stops:list 1.5in'><i>Remark</i>:
      measures have the same scaling as H<span class=GramE>^{</span>-1/2}, but
      neither space includes the other.&nbsp; (Measures are in H<span
      class=GramE>^{</span>-1/2-\eps} though).</li>
 
  </ul>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l6 level2 lfo17;tab-stops:list 1.0in'>One has GWP for real
      random data whose Fourier coefficients decay like 1/|k| (times a Gaussian
      random variable) [<a href="references.html#Bo1995c">Bo1995c</a>].&nbsp;
      Indeed one has an invariant measure.</li>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l6 level2 lfo17;tab-stops:list 1.0in'><span class=SpellE>Solitons</span>
      are asymptotically H^1 stable [MtMe-p3], [<span class=SpellE>MtMe</span>-p].<span
      style='mso-spacerun:yes'>  </span>Indeed, the solution decouples into a
      finite sum of <span class=SpellE>solitons</span> plus dispersive
      radiation [<a href="references.html#EckShr1988">EckShr1988</a>]</li>
 
</ul>
</ul>
 
<div class=MsoNormal align=center style='text-align:center'>
 
<hr size=2 width="100%" align=center>
 
</div>
 
<p class=MsoNormal align=center style='text-align:center'><a name=mkdv></a><b>The
modified <span class=SpellE>KdV</span> equation</b></p>
 
<p>The (<span class=SpellE>defocussing</span>) <span class=SpellE>mKdV</span>
 
equation is </p>
 
<p align=center style='text-align:center'><span class=SpellE>u_t</span> + <span
class=SpellE>u_xxx</span> = 6 u^2 <span class=SpellE>u_x</span>.</p>
 
<p>It is completely <span class=SpellE>integrable</span>, and has infinitely
many conserved quantities.&nbsp; Indeed, for each non-negative integer k, there
is a conserved quantity which is roughly equivalent to the <span class=SpellE>H^k</span>
 
norm of u.&nbsp; This equation has been studied on the <a href="#mKdV_on_R">line</a>,
<a href="#mKdV_on_T">circle</a>, and <a href="#gKdV_on_R+">half-line</a>. </p>
 
<p>The <i>Miura transformation</i> v = <span class=SpellE>u_x</span> + u^2
transforms a solution of <span class=SpellE>defocussing</span> <span
class=SpellE>mKdV</span> to a solution of <a href="#kdv"><span class=SpellE>KdV</span></a>
 
</p>
 
<p align=center style='text-align:center'><span class=SpellE>v_t</span> + <span
class=SpellE>v_xxx</span> = 6 v <span class=SpellE>v_x</span>.</p>
 
<p>Thus one expects the LWP and GWP theory for <span class=SpellE>mKdV</span>
to be one derivative higher than that for <span class=SpellE>KdV</span>. </p>
 
<p>The <span class=SpellE>focussing</span> <span class=SpellE>mKdV</span> </p>
 
<p class=MsoNormal align=center style='text-align:center'><span class=SpellE>u_t</span>
+ <span class=SpellE>u_xxx</span> = - 6 u^2 <span class=SpellE>u_x</span></p>
 
<p><span class=GramE>is</span> very similar, except that the Miura transform is
now v = <span class=SpellE>u_x</span> + <span class=SpellE>i</span> u^2.&nbsp;
This transforms <span class=SpellE>focussing</span> <span class=SpellE>mKdV</span>
to <i>complex-valued</i> <span class=SpellE>KdV</span>, which is a slightly
less tractable equation.&nbsp; (However, the transformed solution v is still
real in the highest order term, so in principle the real-valued theory carries
over to this case). </p>
 
<p>The Miura transformation can be generalized.&nbsp; If v and w solve the
system </p>
 
<p class=MsoNormal align=center style='text-align:center'><span class=SpellE>v_t</span>
+ <span class=SpellE>v_xxx</span> = 6(v^2 + w) <span class=SpellE>v_x</span> <br>
<span class=SpellE>w_t</span> + <span class=SpellE>w_xxx</span> = 6(v^2 + w) <span
class=SpellE>w_x</span></p>
 
<p class=MsoNormal>Then u = v^2 + <span class=SpellE>v_x</span> + w is a
solution of <span class=SpellE>KdV</span>.&nbsp; In particular, if a and b are
constants and v solves </p>
 
<p class=MsoNormal align=center style='text-align:center'><span class=SpellE>v_t</span>
+ <span class=SpellE>v_xxx</span> = 6(a^2 v^2 + <span class=SpellE><span
class=GramE>bv</span></span>) <span class=SpellE>v_x</span></p>
 
<p class=MsoNormal><span class=GramE>then</span> u = a^2 v^2 + <span
class=SpellE>av_x</span> + <span class=SpellE>bv</span> solves <span
class=SpellE>KdV</span> (this is the <i>Gardener transform</i>). </p>
 
<div class=MsoNormal align=center style='text-align:center'>
 
<hr size=2 width="100%" align=center>
 
</div>
 
<p class=MsoNormal align=center style='text-align:center'><a name="mKdV_on_R"></a><span
class=SpellE><span class=GramE><b>mKdV</b></span></span><b> on R and R^+</b></p>
 
<ul type=disc>
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l18 level1 lfo18;tab-stops:list .5in'>Scaling is <span
    class=SpellE>s_c</span> = -1/2.</li>
 
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l18 level1 lfo18;tab-stops:list .5in'>LWP in <span class=SpellE>H^s</span>
    for s &gt;= 1/4 [<a href="references.html#KnPoVe1993">KnPoVe1993</a>]</li>
<ul type=circle>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l18 level2 lfo18;tab-stops:list 1.0in'>Was shown for
      s&gt;3/2 in [<a href="references.html#GiTs1989">GiTs1989</a>]</li>
 
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l18 level2 lfo18;tab-stops:list 1.0in'>This is sharp in the
      <span class=SpellE>focussing</span> case [<a
      href="references.html#KnPoVe-p"><span class=SpellE>KnPoVe</span>-p</a>],
      in the sense that the solution map is no longer uniformly continuous for
      s &lt; 1/4.</li>
  <ul type=square>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l18 level3 lfo18;tab-stops:list 1.5in'>This has been
      extended to the <span class=SpellE>defocussing</span> case in [<span
      class=SpellE>CtCoTa</span>-p], by a high-frequency approximation of <span
      class=SpellE>mKdV</span> by <a href="schrodinger.html#Cubic NLS on R">NLS</a>.&nbsp;
 
      (This high frequency approximation has also been utilized in [<a
      href="references.html#Sch1998">Sch1998</a>]).</li>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l18 level3 lfo18;tab-stops:list 1.5in'>Below 1/4 the
      solution map was known to not be C^3 in [<a
      href="references.html#Bo1993b">Bo1993b</a>], [<a
      href="references.html#Bo1997">Bo1997</a>].</li>
  </ul>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l18 level2 lfo18;tab-stops:list 1.0in'>The same result has
      also been established for the half-line [<span class=SpellE>CoKe</span>-p],
      assuming boundary data is in H<span class=GramE>^{</span>(s+1)/3} of
      course.</li>
 
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l18 level2 lfo18;tab-stops:list 1.0in'>Global weak
      solutions in L^2 were constructed in [<a href="references.html#Ka1983">Ka1983</a>].&nbsp;
      Thus in L^2 one has global existence but no uniform continuity.</li>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l18 level2 lfo18;tab-stops:list 1.0in'>Uniqueness is also
      known when the initial data lies in the weighted space &lt;x&gt;^{3/8}
      u_0 in L^2 [<a href="references.html#GiTs1989">GiTs1989</a>]</li>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l18 level2 lfo18;tab-stops:list 1.0in'>LWP has also been
      demonstrated when &lt;xi&gt;^s hat(u_0) lies in L^{r&#8217;} for 4/3 &lt;
 
      r &lt;= 2 and s &gt;= ½ - 1/2r [Gr-p4]</li>
</ul>
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l18 level1 lfo18;tab-stops:list .5in'>GWP in <span class=SpellE>H^s</span>
    for s &gt; 1/4 [<a href="references.html#CoKeStTaTk-p2">CoKeStTkTa-p2</a>],
    via the <span class=SpellE>KdV</span> theory and the Miura transform, for
    both the <span class=SpellE>focussing</span> and <span class=SpellE>defocussing</span>
 
    cases.</li>
<ul type=circle>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l18 level2 lfo18;tab-stops:list 1.0in'>Was proven for
      s&gt;3/5 in [<a href="references.html#FoLiPo1999">FoLiPo1999</a>]</li>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l18 level2 lfo18;tab-stops:list 1.0in'>Is implicit for s
      &gt;= 1 from [<a href="references.html#KnPoVe1993">KnPoVe1993</a>]</li>
 
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l18 level2 lfo18;tab-stops:list 1.0in'>On the half-line GWP
      is known when s &gt;= 1 and the boundary data is in H^{11/12}, assuming
      compatibility and small L^2 norm [<span class=SpellE>CoKe</span>-p]</li>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l18 level2 lfo18;tab-stops:list 1.0in'>GWP for smooth data
      can also be achieved from inverse scattering methods [<span class=SpellE>BdmFsShp</span>-p];
      the same approach also works on an interval [<span class=SpellE>BdmShp</span>-p].</li>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l18 level2 lfo18;tab-stops:list 1.0in'><span class=SpellE>Solitions</span>
      are asymptotically H^1 stable [MtMe-p3], [<span class=SpellE>MtMe</span>-p]</li>
 
</ul>
</ul>
 
<div class=MsoNormal align=center style='text-align:center'>
 
<hr size=2 width="100%" align=center>
 
</div>
 
<p class=MsoNormal align=center style='text-align:center'><a name="mKdV_on_T"></a><span
class=SpellE><span class=GramE><b>mKdV</b></span></span><b> on T</b></p>
 
<ul type=disc>
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l7 level1 lfo19;tab-stops:list .5in'>Scaling is <span
    class=SpellE>s_c</span> = -1/2.</li>
 
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l7 level1 lfo19;tab-stops:list .5in'>C^0 LWP in L^2 in the
    defocusing case [KpTp-p2]</li>
<ul type=circle>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l7 level2 lfo19;tab-stops:list 1.0in'>C^0 LWP in <span
      class=SpellE>H^s</span> for s &gt; 3/8 [Takaoka and <span class=SpellE>Tsutsumi</span>?]<span
      style='mso-spacerun:yes'>  </span>Note one has to gauge away a nonlinear
      resonance term before one can apply iteration methods.</li>
 
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l7 level2 lfo19;tab-stops:list 1.0in'>Analytic LWP in <span
      class=SpellE>H^s</span> for s &gt;= 1/2, in both focusing and defocusing
      cases [<a href="references.html#KnPoVe1993">KnPoVe1993</a>], [<a
      href="references.html#Bo1993b">Bo1993b</a>].</li>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l7 level2 lfo19;tab-stops:list 1.0in'>This is sharp in the
      sense of analytic well-<span class=SpellE>posedness</span> [<a
      href="references.html#KnPoVe1996">KnPoVe1996</a>] or uniform well-<span
      class=SpellE>posedness</span> [<span class=SpellE>CtCoTa</span>-p]</li>
 
</ul>
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l7 level1 lfo19;tab-stops:list .5in'>C^0 GWP in L^2 in the
    defocusing case [KpTp-p2]</li>
<ul type=circle>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l7 level2 lfo19;tab-stops:list 1.0in'>Analytic GWP in <span
      class=SpellE>H^s</span> for s &gt;= 1/2<span class=GramE>&nbsp; [</span><a
      href="references.html#CoKeStTaTk-p2">CoKeStTkTa-p2</a>], via the <span
      class=SpellE>KdV</span> theory and the Miura transform, for both the <span
      class=SpellE>focussing</span> and <span class=SpellE>defocussing</span>
 
      cases.</li>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l7 level2 lfo19;tab-stops:list 1.0in'>Was proven for s
      &gt;= 1 in [<a href="references.html#KnPoVe1993">KnPoVe1993</a>], [<a
      href="references.html#Bo1993b">Bo1993b</a>].</li>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l7 level2 lfo19;tab-stops:list 1.0in'>One has GWP for
      random data whose Fourier coefficients decay like 1/|k| (times a Gaussian
      random variable) [<a href="references.html#Bo1995c">Bo1995c</a>].&nbsp;
      Indeed one has an invariant measure.&nbsp; Note that such data barely
      fails to be in H<span class=GramE>^{</span>1/2}, however one can modify
      the local well-<span class=SpellE>posedness</span> theory to go below
      H^{1/2} provided that one has a decay like O(|k|^{-1+\eps}) on the
      Fourier coefficients (which is indeed the case almost surely).</li>
 
</ul>
</ul>
 
<div class=MsoNormal align=center style='text-align:center'>
 
<hr size=2 width="100%" align=center>
 
</div>
 
<p class=MsoNormal align=center style='text-align:center'><a name="gKdV_3_on_R"></a><span
class=GramE><b>gKdV_3</b></span><b> on R and R^+</b></p>
 
<ul type=disc>
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l30 level1 lfo20;tab-stops:list .5in'>Scaling is <span
    class=SpellE>s_c</span> = -1/6.</li>
 
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l30 level1 lfo20;tab-stops:list .5in'>LWP in <span class=SpellE>H^s</span>
    for s &gt; -1/6 [Gr-p3]</li>
<ul type=circle>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l30 level2 lfo20;tab-stops:list 1.0in'>Was shown for
      s&gt;=1/12 [<a href="references.html#KnPoVe1993">KnPoVe1993</a>]</li>
 
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l30 level2 lfo20;tab-stops:list 1.0in'>Was shown for
      s&gt;3/2 in [<a href="references.html#GiTs1989">GiTs1989</a>]</li>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l30 level2 lfo20;tab-stops:list 1.0in'>The result s &gt;=
      1/12 has also been established for the half-line [<span class=SpellE>CoKe</span>-p],
      assuming boundary data is in H<span class=GramE>^{</span>(s+1)/3} of
      course..</li>
</ul>
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l30 level1 lfo20;tab-stops:list .5in'>GWP in <span class=SpellE>H^s</span>
 
    for s &gt;= 0 [Gr-p3]</li>
<ul type=circle>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l30 level2 lfo20;tab-stops:list 1.0in'>For s&gt;=1 this is
      in [<a href="references.html#KnPoVe1993">KnPoVe1993</a>]</li>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l30 level2 lfo20;tab-stops:list 1.0in'>Presumably one can
      use either the Fourier truncation method or the &quot;I-method&quot; to
      go below L^2.&nbsp; Even though the equation is not completely <span
      class=SpellE>integrable</span>, the one-dimensional nature of the
      equation suggests that &quot;correction term&quot; techniques will also
      be quite effective.</li>
 
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l30 level2 lfo20;tab-stops:list 1.0in'>On the half-line GWP
      is known when s &gt;= 1 and the boundary data is in H^{5/4}, assuming
      compatibility and small L^2 norm [<span class=SpellE>CoKe</span>-p]</li>
</ul>
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l30 level1 lfo20;tab-stops:list .5in'><span class=SpellE>Solitons</span>
    are H^1-stable [<a href="references.html#CaLo1982">CaLo1982</a>], [<a
    href="references.html#Ws1986">Ws1986</a>], [<a
    href="references.html#BnSouSr1987">BnSouSr1987</a>] and asymptotically H^1
    stable [MtMe-p3], [<span class=SpellE>MtMe</span>-p]</li>
 
</ul>
 
<div class=MsoNormal align=center style='text-align:center'>
 
<hr size=2 width="100%" align=center>
 
</div>
 
<p class=MsoNormal align=center style='text-align:center'><a name="gKdV_3_on_T"></a><span
class=GramE><b>gKdV_3</b></span><b> on T</b></p>
 
<ul type=disc>
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l24 level1 lfo21;tab-stops:list .5in'>Scaling is <span
    class=SpellE>s_c</span> = -1/6.</li>
 
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l24 level1 lfo21;tab-stops:list .5in'>LWP in <span class=SpellE>H^s</span>
    for s&gt;=1/2&nbsp; [<a href="references.html#CoKeStTaTk-p3">CoKeStTkTa-p3</a>]</li>
<ul type=circle>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l24 level2 lfo21;tab-stops:list 1.0in'>Was shown for s
      &gt;= 1 in [<a href="references.html#St1997c">St1997c</a>]</li>
 
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l24 level2 lfo21;tab-stops:list 1.0in'>One has analytic
      ill-<span class=SpellE>posedness</span> for s&lt;1/2 [<a
      href="references.html#CoKeStTaTk-p3">CoKeStTkTa-p3</a>] by a modification
      of the example in [<a href="references.html#KnPoVe1996">KnPoVe1996</a>].</li>
</ul>
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l24 level1 lfo21;tab-stops:list .5in'>GWP in <span class=SpellE>H^s</span>
 
    for s&gt;5/6&nbsp; [<a href="references.html#CoKeStTaTk-p3">CoKeStTkTa-p3</a>]</li>
<ul type=circle>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l24 level2 lfo21;tab-stops:list 1.0in'>Was shown for s
      &gt;= 1 in [<a href="references.html#St1997c">St1997c</a>]</li>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l24 level2 lfo21;tab-stops:list 1.0in'>This result may well
      be improvable by the &quot;damping correction term&quot; method in<span
      class=GramE>&nbsp; [</span><a href="references.html#CoKeStTaTk-p2">CoKeStTkTa-p2</a>].</li>
 
</ul>
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l24 level1 lfo21;tab-stops:list .5in'><i>Remark</i>: For this equation
    it is convenient to make a &quot;gauge transformation'' to subtract off
    the mean of <span class=GramE>P(</span>u).</li>
</ul>
 
<div class=MsoNormal align=center style='text-align:center'>
 
<hr size=2 width="100%" align=center>
 
</div>
 
<p class=MsoNormal align=center style='text-align:center'><a name="gKdV_4_on_R"></a><span
class=GramE><b>gKdV_4</b></span><b> on R and R^+</b></p>
 
<p class=MsoNormal>(Thanks to Felipe <span class=SpellE>Linares</span> for help
with the references here - Ed.)<span style='mso-spacerun:yes'>  </span>A good
survey for the results here is in [Tz-p2].</p>
 
<ul type=disc>
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l22 level1 lfo22;tab-stops:list .5in'>Scaling is <span
    class=SpellE>s_c</span> = 0 (i.e. L^2-critical).</li>
 
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l22 level1 lfo22;tab-stops:list .5in'>LWP in <span class=SpellE>H^s</span>
    for s &gt;= 0 [<a href="references.html#KnPoVe1993">KnPoVe1993</a>]</li>
<ul type=circle>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l22 level2 lfo22;tab-stops:list 1.0in'>Was shown for
      s&gt;3/2 in [<a href="references.html#GiTs1989">GiTs1989</a>]</li>
 
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l22 level2 lfo22;tab-stops:list 1.0in'>The same result s
      &gt;= 0 has also been established for the half-line [<span class=SpellE>CoKe</span>-p],
      assuming boundary data is in H<span class=GramE>^{</span>(s+1)/3} of
      course..</li>
</ul>
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l22 level1 lfo22;tab-stops:list .5in'>GWP in <span class=SpellE>H^s</span>
    for s &gt; 3/4 in both the focusing and defocusing cases, though one must
    of course have smaller L^2 mass than the ground state in the focusing case
    [<span class=SpellE>FoLiPo</span>-p].</li>
 
<ul type=circle>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l22 level2 lfo22;tab-stops:list 1.0in'>For s &gt;= 1 and
      the defocusing case this is in [<a href="references.html#KnPoVe1993">KnPoVe1993</a>]</li>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l22 level2 lfo22;tab-stops:list 1.0in'>Blowup has recently
      been shown for the <span class=SpellE>focussing</span> case for data
      close to a ground state with negative energy [Me-p].&nbsp;&nbsp; In such
      a case the blowup profile must approach the ground state (modulo <span
      class=SpellE>scalings</span> and translations), see [MtMe-p4], [<a
      href="references.html#MtMe2001">MtMe2001</a>].&nbsp; Also, the blow up
      rate in H^1 must be strictly faster than t<span class=GramE>^{</span>-1/3}
      [MtMe-p4], which is the rate suggested by scaling.</li>
 
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l22 level2 lfo22;tab-stops:list 1.0in'>Explicit
      self-similar blow-up solutions have been constructed [<span class=SpellE>BnWe</span>-p]
      but these are not in L^2.</li>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l22 level2 lfo22;tab-stops:list 1.0in'>GWP for small L^2
      data in either case [<a href="references.html#KnPoVe1993">KnPoVe1993</a>].&nbsp;
      In the <span class=SpellE>focussing</span> case we have GWP whenever the
      L^2 norm is strictly smaller than that of the ground state Q (thanks to
      Weinstein's sharp <span class=SpellE>Gagliardo-Nirenberg</span>
 
      inequality).&nbsp; It seems like a reasonable (but difficult) conjecture
      to have GWP for large L^2 data in the defocusing case.</li>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l22 level2 lfo22;tab-stops:list 1.0in'>On the half-line GWP
      is known when s &gt;= 1 and the boundary data is in H^{11/12}, assuming
      compatibility and small L^2 norm [<span class=SpellE>CoKe</span>-p]</li>
</ul>
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l22 level1 lfo22;tab-stops:list .5in'><span class=SpellE>Solitons</span>
    are H^1-unstable [<a href="references.html#MtMe2001">MtMe2001</a>].&nbsp;&nbsp;
 
    However, small H^1 perturbations of a <span class=SpellE>soliton</span>
    must asymptotically converge weakly to some rescaled <span class=SpellE>soliton</span>
    shape provided that the H^1 norm stays comparable to 1 [<a
    href="references.html#MtMe-p"><span class=SpellE>MtMe</span>-p</a>].</li>
</ul>
 
<div class=MsoNormal align=center style='text-align:center'>
 
<hr size=2 width="100%" align=center>
 
</div>
 
<p class=MsoNormal align=center style='text-align:center'><a name="gKdV_4_on_T"></a><span
class=GramE><b>gKdV_4</b></span><b> on T</b></p>
 
<ul type=disc>
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l14 level1 lfo23;tab-stops:list .5in'>Scaling is <span
    class=SpellE>s_c</span> = 0.</li>
 
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l14 level1 lfo23;tab-stops:list .5in'>LWP in <span class=SpellE>H^s</span>
    for s&gt;=1/2&nbsp; [<a href="references.html#CoKeStTaTk-p3">CoKeStTkTa-p3</a>]</li>
<ul type=circle>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l14 level2 lfo23;tab-stops:list 1.0in'>Was shown for s
      &gt;= 1 in [<a href="references.html#St1997c">St1997c</a>]</li>
 
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l14 level2 lfo23;tab-stops:list 1.0in'>Analytic well-<span
      class=SpellE>posedness</span> fails for s &lt; 1/2; this is essentially
      in [<a href="references.html#KnPoVe1996">KnPoVe1996</a>]</li>
</ul>
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l14 level1 lfo23;tab-stops:list .5in'>GWP in <span class=SpellE>H^s</span>
    for s&gt;=1 [<a href="references.html#St1997c">St1997c</a>]</li>
 
<ul type=circle>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l14 level2 lfo23;tab-stops:list 1.0in'>This is almost
      certainly improvable by the techniques in [<a
      href="references.html#CoKeStTaTk-p3">CoKeStTkTa-p3</a>], probably to s
      &gt; 6/7.&nbsp; There are some low-frequency issues which may require the
      techniques in [<a href="references.html#KeTa-p"><span class=SpellE>KeTa</span>-p</a>].</li>
</ul>
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l14 level1 lfo23;tab-stops:list .5in'><i>Remark</i>: For this
    equation it is convenient to make a &quot;gauge transformation'' to
    subtract off the mean of <span class=GramE>P(</span>u).</li>
 
</ul>
 
<div class=MsoNormal align=center style='text-align:center'>
 
<hr size=2 width="100%" align=center>
 
</div>
 
<p class=MsoNormal align=center style='text-align:center'><a name="gKdV_on_R+"></a><span
class=SpellE><span class=GramE><b>gKdV</b></span></span><b> on R^+</b></p>
 
<ul type=disc>
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l19 level1 lfo24;tab-stops:list .5in'>The <span class=SpellE>gKdV</span>
 
    Cauchy-boundary problem on the half-line is</li>
</ul>
 
<p class=MsoNormal align=center style='text-align:center'><span class=SpellE>u_t</span>
+ u<span class=GramE>_{</span>xxx} + <span class=SpellE>u_x</span> + <span
class=SpellE>u^k</span> <span class=SpellE>u_x</span> = 0;&nbsp; u(x,0) =
u_0(x);&nbsp; u(0,t) = h(t)</p>
 
<p class=MsoNormal style='margin-left:.5in'>The sign of u<span class=GramE>_{</span>xxx}
is important (it makes the influence of the boundary x=0 mostly negligible),
the sign of u <span class=SpellE>u_x</span> is not.&nbsp; The drift term <span
class=SpellE>u_x</span> is convenient for technical reasons; it is not known
whether it is truly necessary. </p>
 
<ul type=disc>
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l9 level1 lfo25;tab-stops:list .5in'>LWP is known for initial
    data in <span class=SpellE>H^s</span> and boundary data in H<span
    class=GramE>^{</span>(s+1)/3} when s &gt; 3/4 [<span class=SpellE>CoKn</span>-p].</li>
 
<ul type=circle>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l9 level2 lfo25;tab-stops:list 1.0in'>The techniques are
      based on [<a href="references.html#KnPoVe1993">KnPoVe1993</a>] and a
      replacement of the IVBP with a forced IVP.</li>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l9 level2 lfo25;tab-stops:list 1.0in'>This has been
      improved to s &gt;= <span class=SpellE>s_c</span> = 1/2 - 2/k when k &gt;
      4 [<span class=SpellE>CoKe</span>-p].</li>
 
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l9 level2 lfo25;tab-stops:list 1.0in'>For <a
      href="#KdV_on_R+"><span class=SpellE>KdV</span></a>, <a href="#mKdV_on_R"><span
      class=SpellE>mKdV</span></a>, <a href="#gKdV_3_on_R">gKdV-3</a> , and <a
      href="#gKdV_4_on_R">gKdV-4</a> see the corresponding sections on this
      page.</li>
</ul>
</ul>
 
<div class=MsoNormal align=center style='text-align:center'>
 
<hr size=2 width="100%" align=center>
 
</div>
 
<p align=center style='text-align:center'><a name="Schrodinger_Airy"></a><b>Nonlinear
<span class=SpellE>Schrodinger</span>-Airy equation<o:p></o:p></b></p>
 
<p>The equation</p>
 
<p align=center style='text-align:center'><span class=SpellE>u_t</span> + <span
class=SpellE>i</span> c <span class=SpellE>u_xx</span> + <span class=SpellE>u_xxx</span>
 
= <span class=SpellE>i</span> gamma |u|^2 u + delta |u|^2 <span class=SpellE>u_x</span>
+ epsilon u^2 <span class=SpellE><u>u</u>_x</span></p>
 
<p><span class=GramE>on</span> R is a combination of the <a
href="schrodinger.html#Cubic_NLS_on_R">cubic NLS equation</a> , the <a
href="schrodinger.html#dnls-3_on_R">derivative cubic NLS equation</a>, <a
href="#mKdV_on_R">complex <span class=SpellE>mKdV</span></a>, and a cubic
nonlinear Airy equation.<span style='mso-spacerun:yes'>  </span>This equation
is a general model for <span class=SpellE>propogation</span> of pulses in an
optical fiber [<a href="references.html#Kod1985">Kod1985</a>], [<a
href="references.html#HasKod1987">HasKod1987</a>]</p>
 
<p style='margin-left:.5in;text-indent:-.25in;mso-list:l25 level1 lfo26;
tab-stops:list .5in'><![if !supportLists]><span style='font-family:Symbol;
mso-fareast-font-family:Symbol;mso-bidi-font-family:Symbol'><span
style='mso-list:Ignore'>·<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
</span></span></span><![endif]>When c=delta=epsilon = 0, scaling is s=-1.<span
style='mso-spacerun:yes'>  </span>When c=gamma=0, scaling is &#8211;1/2.</p>
 
<p style='margin-left:.5in;text-indent:-.25in;mso-list:l25 level1 lfo26;
tab-stops:list .5in'><![if !supportLists]><span style='font-family:Symbol;
mso-fareast-font-family:Symbol;mso-bidi-font-family:Symbol'><span
style='mso-list:Ignore'>·<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
</span></span></span><![endif]>LWP is known when s &gt;= ¼ [<a
href="references.html#St1997d">St1997d</a>]</p>
 
<p style='margin-left:1.0in;text-indent:-.25in;mso-list:l25 level2 lfo26;
tab-stops:list 1.0in'><![if !supportLists]><span style='font-family:"Courier New";
mso-fareast-font-family:"Courier New"'><span style='mso-list:Ignore'>o<span
style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]>For
s &gt; ¾ this is in [<a href="references.html#Lau1997">Lau1997</a>], [<a
href="references.html#Lau2001">Lau2001</a>]</p>
 
<p style='margin-left:1.0in;text-indent:-.25in;mso-list:l25 level2 lfo26;
tab-stops:list 1.0in'><![if !supportLists]><span style='font-family:"Courier New";
mso-fareast-font-family:"Courier New"'><span style='mso-list:Ignore'>o<span
style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]>The
s&gt;=1/4 result is also known when c is a time-dependent function [Cv2002],
[CvLi2003]</p>
 
<p style='margin-left:1.0in;text-indent:-.25in;mso-list:l25 level2 lfo26;
tab-stops:list 1.0in'><![if !supportLists]><span style='font-family:"Courier New";
mso-fareast-font-family:"Courier New"'><span style='mso-list:Ignore'>o<span
style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]>For
s &lt; -1/4 and delta or epsilon non-zero, the solution map is not C^3 [<span
class=SpellE>CvLi</span>-p]</p>
 
<p style='margin-left:1.0in;text-indent:-.25in;mso-list:l25 level2 lfo26;
tab-stops:list 1.0in'><![if !supportLists]><span style='font-family:"Courier New";
mso-fareast-font-family:"Courier New"'><span style='mso-list:Ignore'>o<span
style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]>When
delta = epsilon = 0 LWP is known for s &gt; -1/4 [<a
href="references.html#Cv2004">Cv2004</a>]</p>
 
<p style='margin-left:1.5in;text-indent:-.25in;mso-list:l25 level3 lfo26;
tab-stops:list 1.5in'><![if !supportLists]><span style='font-family:Wingdings;
mso-fareast-font-family:Wingdings;mso-bidi-font-family:Wingdings'><span
style='mso-list:Ignore'>§<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
</span></span></span><![endif]>For s &lt; -1/4 the solution map is not C^3 [<span
class=SpellE>CvLi</span>-p]</p>
 
<div class=MsoNormal align=center style='text-align:center'>
 
<hr size=2 width="100%" align=center>
 
</div>
 
<p class=MsoNormal align=center style='text-align:center'><a name=gKdV></a><b>Miscellaneous
 
<span class=SpellE>gKdV</span> results</b></p>
 
<p class=MsoNormal>[Thanks to <span class=SpellE>Nikolaos</span> <span
class=SpellE>Tzirakis</span> for some corrections - Ed.] </p>
 
<ul type=disc>
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l2 level1 lfo27;tab-stops:list .5in'>On R with k &gt; 4, <span
    class=SpellE>gKdV</span>-k is LWP down to scaling: s &gt;= <span
    class=SpellE>s_c</span> = 1/2 - 2/k [<a href="references.html#KnPoVe1993">KnPoVe1993</a>]</li>
 
<ul type=circle>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l2 level2 lfo27;tab-stops:list 1.0in'>Was shown for
      s&gt;3/2 in [<a href="references.html#GiTs1989">GiTs1989</a>]</li>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l2 level2 lfo27;tab-stops:list 1.0in'>One has ill-<span
      class=SpellE>posedness</span> in the supercritical regime [<a
      href="references.html#BirKnPoSvVe1996">BirKnPoSvVe1996</a>]</li>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l2 level2 lfo27;tab-stops:list 1.0in'>For small data one
      has scattering [<a href="references.html#KnPoVe1993c">KnPoVe1993c</a>].<span
      style='mso-spacerun:yes'>  </span>Note that one cannot have scattering in
      L^2 except in the critical case k=4 because one can scale <span
      class=SpellE>solitons</span> to be arbitrarily small in the non-critical
      cases.</li>
 
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l2 level2 lfo27;tab-stops:list 1.0in'><span class=SpellE>Solitons</span>
      are H^1-unstable [<a href="references.html#BnSouSr1987">BnSouSr1987</a>]</li>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l2 level2 lfo27;tab-stops:list 1.0in'>If one considers an
      arbitrary smooth non-linearity (not necessarily a power) then one has LWP
      for small data in <span class=SpellE>H^s</span>, s &gt; 1/2 [<a
      href="references.html#St1995">St1995</a>]</li>
 
</ul>
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l2 level1 lfo27;tab-stops:list .5in'>On R with any k, <span
    class=SpellE>gKdV</span>-k is GWP in <span class=SpellE>H^s</span> for s
    &gt;= 1 [<a href="references.html#KnPoVe1993">KnPoVe1993</a>], though for
    k &gt;= 4 one needs the L^2 norm to be small; global weak solutions were
    constructed much earlier, with the same smallness assumption when k &gt;=
    4.&nbsp; This should be improvable below H^1 for all k.</li>
 
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l2 level1 lfo27;tab-stops:list .5in'>On R with any k, <span
    class=SpellE>gKdV</span>-k has the <span class=SpellE>H^s</span> norm
    growing like t^{(s-1)+} in time for any integer s &gt;= 1 [<a
    href="references.html#St1997b">St1997b</a>]</li>
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l2 level1 lfo27;tab-stops:list .5in'>On R with any non-linearity,
    a non-zero solution to <span class=SpellE>gKdV</span> cannot be supported
    on the half-line R^+ (or R^-) for two different times [<a
    href="references.html#KnPoVe-p3">KnPoVe-p3</a>], [KnPoVe-p4].</li>
 
<ul type=circle>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l2 level2 lfo27;tab-stops:list 1.0in'>In the completely <span
      class=SpellE>integrable</span> cases k=1,2 this is in [<a
      href="references.html#Zg1992">Zg1992</a>]</li>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l2 level2 lfo27;tab-stops:list 1.0in'>Also, a non-zero
      solution to <span class=SpellE>gKdV</span> cannot vanish on a rectangle
      in <span class=SpellE>spacetime</span> [<a
      href="references.html#SauSc1987">SauSc1987</a>]; see also [<a
      href="references.html#Bo1997b">Bo1997b</a>].</li>
 
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l2 level2 lfo27;tab-stops:list 1.0in'>Extensions to higher
      order <span class=SpellE>gKdV</span> type equations are in [<a
      href="references.html#Bo1997b">Bo1997b</a>], [KnPoVe-p5].</li>
</ul>
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l2 level1 lfo27;tab-stops:list .5in'>On R with non-integer k, one
    has decay of <span class=GramE>O(</span>t^{-1/3}) in L^\<span
    class=SpellE>infty</span> for small decaying data if k &gt; (19 - <span
    class=SpellE>sqrt</span>(57))/4 ~ 2.8625... [<a
    href="references.html#CtWs1991">CtWs1991</a>]</li>
 
<ul type=circle>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l2 level2 lfo27;tab-stops:list 1.0in'>A similar result for
      k &gt; (5+<span class=GramE>sqrt(</span>73))/4 ~ 3.39... <span
      class=GramE>was</span> obtained in [<a href="references.html#PoVe1990">PoVe1990</a>].</li>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l2 level2 lfo27;tab-stops:list 1.0in'>When k=2 solutions
      decay like O(t^{-1/3}), and when k=1 solutions decay generically like
      O(t^{-2/3}) but like O( (t/log t)^{-2/3}) for exceptional data [<a
      href="references.html#AbSe1977">AbSe1977</a>]</li>
 
</ul>
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l2 level1 lfo27;tab-stops:list .5in'>In the L^2 <span
    class=SpellE>subcritical</span> case 0 &lt; k &lt; 4, <span class=SpellE>multisoliton</span>
    solutions are asymptotically H^1-stable [<span class=SpellE>MtMeTsa</span>-p]</li>
 
<ul type=circle>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l2 level2 lfo27;tab-stops:list 1.0in'>For a single <span
      class=SpellE>soliton</span> this is in [MtMe-p3], [<span class=SpellE>MtMe</span>-p],
      [<a href="references.html#Miz2001">Miz2001</a>]; earlier work is in [<a
      href="references.html#Bj1972">Bj1972</a>], [<a
      href="references.html#Bn1975">Bn1975</a>], [<a
      href="references.html#Ws1986">Ws1986</a>], [<a
      href="references.html#PgWs1994">PgWs1994</a>]</li>
 
</ul>
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l2 level1 lfo27;tab-stops:list .5in'>A dissipative version of <span
    class=SpellE>gKdV</span>-k was analyzed in [<a
    href="references.html#MlRi2001">MlRi2001</a>]</li>
</ul>
 
<ul type=disc>
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l20 level1 lfo28;tab-stops:list .5in'>On T with any k, <span
    class=SpellE>gKdV</span>-k has the <span class=SpellE>H^s</span> norm
    growing like t^{2(s-1)+} in time for any integer s &gt;= 1 [<a
    href="references.html#St1997b">St1997b</a>]</li>
 
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l20 level1 lfo28;tab-stops:list .5in'>On T with k &gt;= 3, <span
    class=SpellE>gKdV</span>-k is LWP for s &gt;= 1/2 [<a
    href="references.html#CoKeStTaTk-p3">CoKeStTkTa-p3</a>]</li>
<ul type=circle>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l20 level2 lfo28;tab-stops:list 1.0in'>Was shown for s
      &gt;= 1 in [<a href="references.html#St1997c">St1997c</a>]</li>
 
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l20 level2 lfo28;tab-stops:list 1.0in'>Analytic well-<span
      class=SpellE>posedness</span> fails for s &lt; 1/2 [<a
      href="references.html#CoKeStTaTk-p3">CoKeStTkTa-p3</a>], [<a
      href="references.html#KnPoVe1996">KnPoVe1996</a>]</li>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l20 level2 lfo28;tab-stops:list 1.0in'>For arbitrary smooth
      non-<span class=SpellE>linearities</span>, weak H^1 solutions were
      constructed in [<a href="references.html#Bo1993b">Bo1993</a>].</li>
 
</ul>
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l20 level1 lfo28;tab-stops:list .5in'>On T with k &gt;= 3, <span
    class=SpellE>gKdV</span>-k is GWP for s &gt;= 1 except in the <span
    class=SpellE>focussing</span> case [<a href="references.html#St1997c">St1997c</a>]</li>
<ul type=circle>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l20 level2 lfo28;tab-stops:list 1.0in'>The estimates in [<a
      href="references.html#CoKeStTaTk-p3">CoKeStTkTa-p3</a>] suggest that this
      is improvable to 13/14 - 2/7k, but this has only been proven in the
      sub-critical case k=3 [<a href="references.html#CoKeStTaTk-p3">CoKeStTkTa-p3</a>].&nbsp;
 
      In the critical and super-critical cases there are some low-frequency
      issues which may require the techniques in [<a
      href="references.html#KeTa-p"><span class=SpellE>KeTa</span>-p</a>].</li>
</ul>
</ul>
 
<div class=MsoNormal align=center style='text-align:center'>
 
<hr size=2 width="100%" align=center>
 
</div>
 
<p class=MsoNormal align=center style='text-align:center'><a name=hierarchy></a><b>The
 
<span class=SpellE>KdV</span> Hierarchy</b></p>
 
<p>The <span class=SpellE>KdV</span> equation </p>
 
<p align=center style='text-align:center'><span class=SpellE>V_t</span> + <span
class=SpellE>V_xxx</span> = 6 <span class=SpellE>V_x</span></p>
 
<p><span class=GramE>can</span> be rewritten in the Lax Pair form </p>
 
<p align=center style='text-align:center'><span class=SpellE>L_t</span> = [L,
P]</p>
 
<p><span class=GramE>where</span> L is the second-order operator </p>
 
<p align=center style='text-align:center'>L = -D^2 + V</p>
 
<p>(D = d/<span class=SpellE>dx</span>) and P is the third-order <span
class=SpellE>antiselfadjoint</span> operator </p>
 
<p align=center style='text-align:center'>P = 4D^3 + 3(DV + VD).</p>
 
<p>(<span class=GramE>note</span> that P consists of the <span class=SpellE>zeroth</span>
 
order and higher terms of the formal power series expansion of 4i L^{3/2}). </p>
 
<p>One can replace P with other fractional powers of L.&nbsp; For instance, the
<span class=SpellE>zeroth</span> order and higher terms of 4i L<span
class=GramE>^{</span>5/2} are </p>
 
<p align=center style='text-align:center'>P = 4D^5 + 5(D^3 V + V D^3) - 5/4 (D <span
class=SpellE>V_xx</span> + <span class=SpellE>V_xx</span> D) + 15/4 (D V^2 +
V^2 D)</p>
 
<p><span class=GramE>and</span> the Lax pair equation becomes </p>
 
<p class=MsoNormal align=center style='text-align:center'><span class=SpellE>V_t</span>
+ <span class=SpellE>u_xxxxx</span> = (5 V_x^2 + 10 V <span class=SpellE>V_xx</span>
+ 10 V^3<span class=GramE>)_</span>x</p>
 
<p><span class=GramE>with</span> Hamiltonian </p>
 
<p class=MsoNormal align=center style='text-align:center'><span class=GramE>H(</span>V)
= \<span class=SpellE>int</span>&nbsp; V_xx^2 - 5 V^2 <span class=SpellE>V_xx</span>
- 5 V^4.</p>
 
<p>These flows all commute with each <span class=GramE>other,</span> and their
Hamiltonians are conserved by all the flows simultaneously. </p>
 
<p>The <span class=SpellE>KdV</span> <span class=GramE>hierarchy are</span>
examples of higher order water wave models; a general formulation is</p>
 
<p align=center style='text-align:center'><span class=SpellE>u_t</span> + <span
class=SpellE>partial_x</span><span class=GramE>^{</span>2j+1} u = P(u, <span
class=SpellE>u_x</span>, ..., <span class=SpellE>partial_x</span>^{2j} u)</p>
 
<p><span class=GramE>where</span> u is real-valued and P is a polynomial with
no constant or linear terms; thus <span class=SpellE>KdV</span> and <span
class=SpellE>gKdV</span> correspond to j=1, and the higher order equations in
the hierarchy correspond to j=2,3,etc.<span style='mso-spacerun:yes'>
</span>LWP for these equations in high regularity <span class=SpellE>Sobolev</span>
spaces is in [<a href="references.html#KnPoVe1994">KnPoVe1994</a>], and
independently by <span class=SpellE><span class=GramE>Cai</span></span> (ref?);
see also [<a href="references.html#CrKpSr1992">CrKpSr1992</a>].<span
style='mso-spacerun:yes'>  </span>The case j=2 was studied by <span
class=SpellE>Choi</span> (ref?).<span style='mso-spacerun:yes'>  </span>The
non-scalar diagonal case was treated in [<a href="references.html#KnSt1997">KnSt1997</a>];
the periodic case was studied in [Bo-p3].<span style='mso-spacerun:yes'>
 
</span>Note in the periodic case it is possible to have ill-<span class=SpellE>posedness</span>
for every regularity, for instance <span class=SpellE>u_t</span> + <span
class=SpellE>u_xxx</span> = u^2 u_x^2 is ill-posed in every <span class=SpellE>H^s</span>
[Bo-p3]</p>
 
<p><o:p>&nbsp;</o:p></p>
 
<div class=MsoNormal align=center style='text-align:center'>
 
<hr size=2 width="100%" align=center>
 
</div>
 
<p class=MsoNormal align=center style='text-align:center'><a name=Benjamin-Ono></a><b>Benjamin-Ono
equation</b></p>
 
<p class=MsoNormal>[Thanks to <span class=SpellE>Nikolay</span> <span
class=SpellE>Tzvetkov</span> and Felipe <span class=SpellE>Linares</span> for
help with this section - Ed] </p>
 
<p>The <i>generalized Benjamin-Ono equation</i> <span class=SpellE>BO_a</span>
is the scalar equation </p>
 
<p class=MsoNormal align=center style='text-align:center'><span class=SpellE>u_t</span>
+ <span class=SpellE>D_x</span><span class=GramE>^{</span>1+a} <span
class=SpellE>u_x</span> + <span class=SpellE>uu_x</span> = 0</p>
 
<p class=MsoNormal><span class=GramE>where</span> <span class=SpellE>D_x</span>
= <span class=SpellE>sqrt</span>{-Delta} is the positive differentiation
operator.&nbsp; When a=1 this is <a href="#kdv"><span class=SpellE>KdV</span></a>;
when a=0 this is the Benjamin-Ono equation (BO) [<a
href="references.html#Bj1967">Bj1967</a>], [<a href="references.html#On1975">On1975</a>],
which models one-dimensional internal waves in deep water.&nbsp; Both of these
equations are completely <span class=SpellE>integrable</span> (see e.g. [<a
href="references.html#AbFs1983">AbFs1983</a>], [<a
href="references.html#CoiWic1990">CoiWic1990</a>]), though the intermediate
cases 0 &lt; a &lt; 1 are not. </p>
 
<p>When a=0, scaling is s = -1/2, and the following results are known: </p>
 
<ul type=disc>
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l26 level1 lfo29;tab-stops:list .5in'>LWP in <span class=SpellE>H^s</span>
    for s &gt;= 1 [Ta-p]</li>
<ul type=circle>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l26 level2 lfo29;tab-stops:list 1.0in'>For s &gt;= 9/8 this
      is in [<span class=SpellE>KnKoe</span>-p]</li>
 
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l26 level2 lfo29;tab-stops:list 1.0in'>For s &gt;= 5/4 this
      is in [<span class=SpellE>KocTz</span>-p]</li>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l26 level2 lfo29;tab-stops:list 1.0in'>For s &gt;= 3/2 this
      is in [<a href="references.html#Po1991">Po1991</a>]</li>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l26 level2 lfo29;tab-stops:list 1.0in'>For s &gt; 3/2 this
      is in [<a href="references.html#Io1986">Io1986</a>]</li>
 
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l26 level2 lfo29;tab-stops:list 1.0in'>For s &gt; 3 this is
      in [<a href="references.html#Sau1979">Sau1979</a>]</li>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l26 level2 lfo29;tab-stops:list 1.0in'>For no value of s is
      the solution map uniformly continuous [KocTz-p2]</li>
  <ul type=square>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l26 level3 lfo29;tab-stops:list 1.5in'>For s &lt; -1/2
      this is in [<span class=SpellE>BiLi</span>-p]</li>
 
  </ul>
</ul>
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l26 level1 lfo29;tab-stops:list .5in'>Global weak solutions exist
    for L^2 data [<a href="references.html#Sau1979">Sau1979</a>], [<a
    href="references.html#GiVl1989b">GiVl1989b</a>], [<a
    href="references.html#GiVl1991">GiVl1991</a>], [<a
    href="references.html#Tom1990">Tom1990</a>]</li>
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l26 level1 lfo29;tab-stops:list .5in'>Global well-<span
    class=SpellE>posedness</span> in <span class=SpellE>H^s</span> for s &gt;=
    1 [Ta-p]</li>
 
<ul type=circle>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l26 level2 lfo29;tab-stops:list 1.0in'>For s &gt;= 3/2 this
      is in [<a href="references.html#Po1991">Po1991</a>]</li>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l26 level2 lfo29;tab-stops:list 1.0in'>For smooth solutions
      this is in [<a href="references.html#Sau1979">Sau1979</a>]</li>
</ul>
</ul>
 
<p class=MsoNormal>When 0 &lt; a &lt; 1, scaling is s = -1/2 - <span
class=GramE>a,</span> and the following results are known: </p>
 
<ul type=disc>
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l1 level1 lfo30;tab-stops:list .5in'>LWP in <span class=SpellE>H^s</span>
    is known for s &gt; 9/8 &#8211; 3a/8 [<span class=SpellE>KnKoe</span>-p]</li>
 
<ul type=circle>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l1 level2 lfo30;tab-stops:list 1.0in'>For s &gt;= 3/4 (2-a)
      this is in [<a href="references.html#KnPoVe1994b">KnPoVe1994b</a>]</li>
</ul>
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l1 level1 lfo30;tab-stops:list .5in'>GWP is known when s &gt;=
    (a+1)/2 when a &gt; 4/5, from the conservation of the Hamiltonian [<a
    href="references.html#KnPoVe1994b">KnPoVe1994b</a>]</li>
 
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l1 level1 lfo30;tab-stops:list .5in'>The LWP results are obtained
    by energy methods; it is known that pure iteration methods cannot work [<a
    href="references.html#MlSauTz2001">MlSauTz2001</a>]</li>
<ul type=circle>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l1 level2 lfo30;tab-stops:list 1.0in'>However, this can be
      salvaged by combining the <span class=SpellE>H^s</span> norm || f ||_{<span
      class=SpellE>H^s</span>} with a weighted <span class=SpellE>Sobolev</span>
      space, namely || <span class=SpellE>xf</span> ||_{H^{s - 2s_*}}, where
      s_* = (a+1)/2 is the energy regularity.[CoKnSt-p4]</li>
 
</ul>
</ul>
 
<p class=MsoNormal>One can replace the quadratic non-linearity <span
class=SpellE>uu_x</span> by higher powers u<span class=GramE>^{</span>k-1} <span
class=SpellE>u_x</span>, in analogy with <span class=SpellE>KdV</span> and <span
class=SpellE>gKdV</span>, giving rise to the <span class=SpellE>gBO</span>-k
equations (let us take a=0 for sake of discussion).<span
style='mso-spacerun:yes'>  </span>The scaling exponent is 1/2 - 1<span
class=GramE>/(</span>k-1).</p>
 
<ul type=disc>
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l21 level1 lfo31;tab-stops:list .5in'>For k=3, one has GWP for
    large data in H^1 [<span class=SpellE>KnKoe</span>-p] and LWP for small
    data in <span class=SpellE>H^s</span>, s &gt; ½ [<span class=SpellE>MlRi</span>-p]</li>
<ul type=circle>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l21 level2 lfo31;tab-stops:list 1.0in'>For small data in <span
      class=SpellE>H^s</span>, s&gt;1, LWP was obtained in [<a
      href="references.html#KnPoVe1994b">KnPoVe1994b</a>]</li>
 
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l21 level2 lfo31;tab-stops:list 1.0in'>With the addition of
      a small viscosity term, GWP can also be obtained in H^1 by complete <span
      class=SpellE>integrability</span> methods in [FsLu2000], with <span
      class=SpellE>asymptotics</span> under the additional assumption that the
      initial data is in L^1.</li>
  <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:
      auto;mso-list:l21 level2 lfo31;tab-stops:list 1.0in'>For s &lt; ½, the
      solution map is not C^3 [<span class=SpellE>MlRi</span>-p]</li>
</ul>
 
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l21 level1 lfo31;tab-stops:list .5in'>For k=4, LWP for small data
    in <span class=SpellE>H^s</span>, s &gt; 5/6 was obtained in [<a
    href="references.html#KnPoVe1994b">KnPoVe1994b</a>].</li>
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l21 level1 lfo31;tab-stops:list .5in'>For k&gt;4, LWP for small
    data in <span class=SpellE>H^s</span>, s &gt;=3/4 was obtained in [<a
    href="references.html#KnPoVe1994b">KnPoVe1994b</a>].</li>
 
<li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
    mso-list:l21 level1 lfo31;tab-stops:list .5in'>For any k &gt;= 3 and s
    &lt; 1/2 - 1/k the solution map is not uniformly continuous [<span
    class=SpellE>BiLi</span>-p]</li>
</ul>
 
<p class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto'>The
<span class=SpellE>KdV</span>-Benjamin Ono (<span class=SpellE>KdV</span>-BO)
equation is formed by combining the linear parts of the <span class=SpellE>KdV</span>
 
and Benjamin-Ono equations together.<span style='mso-spacerun:yes'>  </span>It
is globally well-posed in L^2 [<a href="references.html#Li1999">Li1999</a>],
and locally well-posed in H<span class=GramE>^{</span>-3/4+} [<span
class=SpellE>KozOgTns</span>] (see also [<span class=SpellE>HuoGuo</span>-p]
where H^{-1/8+} is obtained).<span style='mso-spacerun:yes'>  </span>Similarly
one can generalize the non-linearity to be k-linear, generating for instance
the modified <span class=SpellE>KdV</span>-BO equation, which is locally
well-posed in H<span class=GramE>^{</span>1/4+} [<span class=SpellE>HuoGuo</span>-p].<span
style='mso-spacerun:yes'>  </span>For general <span class=SpellE>gKdV-gBO</span>
 
equations one has local well-<span class=SpellE><span class=GramE>posedness</span></span><span
class=GramE><span style='mso-spacerun:yes'>  </span>in</span> H^3 and above [<a
href="references.html#GuoTan1992">GuoTan1992</a>].<span
style='mso-spacerun:yes'>  </span>One can also add damping terms <span
class=SpellE>Hu_x</span> to the equation; this arises as a model for ion-acoustic
waves of finite amplitude with linear Landau damping [<a
href="references.html#OttSud1982">OttSud1982</a>].</p>
 
</div>
 
</body>
 
</html>

Latest revision as of 15:16, 4 May 2010

Equations of Korteweg-de Vries type

The equations of Korteweg-de Vries type are all nonlinear perturbations of the Airy equation. They take the general form

where is a function of one space and one time variable, and is some polynomial of . One can place various normalizing constants in front of the and terms, but they can usually be scaled out. The function and the polynomial are usually assumed to be real.

The variable is usually assumed to live on the real line R (so there is some decay at infinity) or on the torus T (so the data is periodic). The half-line has also been studied, as has the case of periodic data with large period. It might be interesting to look at whether the periodicity assumption can be perturbed (e.g. quasi-periodic data); it is not clear whether the phenomena we see in the periodic problem are robust under perturbations, or are number-theoretic artefacts of perfect periodicity.

Specific equations

Several special cases of KdV-type equations are of interest, including

Drift terms can be added, but they can be subsumed into the polynomial or eliminated by a Galilean transformation (except in the half-line case). Indeed, one can freely insert or remove any term of the form by shifting the variable by , which is especially useful for periodic higher-order gKdV equations (setting equal to the mean of ).

The Korteweg-de Vries equation is also a member of the KdV hierarchy. One can also couple the KdV equation to other equations, creating for instance the nonlinear Schrodinger-Airy system.

History

Historically, these types of equations first arose in the study of 2D shallow wave propagation, but have since appeared as limiting cases of many dispersive models. Interestingly, the 2D shallow wave equation can also give rise to the Boussinesq or cubic NLS equation by making more limiting assumptions (in particular, weak nonlinearity and slowly varying amplitude).

Conservation laws, symmetries, and criticality

KdV-type equations on R or T always come with three conserved quantities:

where is a primitive of . Note that the Hamiltonian is positive-definite in the defocussing cases (if u is real); thus the defocussing equations have a better chance of long-term existence. The mass has no definite sign and so is only useful in specific cases (e.g. perturbations of a soliton).

In general, the above three quantities are the only conserved quantities available, but the KdV and mKdV equations come with infinitely many more such conserved quantities due to their completely integrable nature.

The critical (or scaling) regularity is

In particular, KdV, mKdV, and gKdV-3 are subcritical with respect to , gKdV-4 is critical, and all the other equations are supercritical. Generally speaking, the potential energy term can be pretty much ignored in the sub-critical equations, needs to be dealt with carefully in the critical equation, and can completely dominate the Hamiltonian in the super-critical equations (to the point that blowup occurs if the equation is not defocussing. Note that is always a sub-critical regularity.

The dispersion relation is always increasing, which means that singularities always propagate to the left. In fact, high frequencies propagate leftward at extremely high speeds, which causes a smoothing effect if there is some decay in the initial data ( will do). On the other hand, KdV-type equations have the remarkable property of supporting localized travelling wave solutions known as solitons, which propagate to the right. It is known that solutions to the completely integrable equations (i.e. KdV and mKdV always resolve to a superposition of solitons as , but it is an interesting open question as to whether the same phenomenon occurs for the other KdV-type equations.

Symplectic structure

A KdV-type equation can be viewed as a symplectic flow with the Hamiltonian defined above, and the symplectic form given by

.

Thus is the natural Hilbert space in which to study the symplectic geometry of these flows. Unfortunately, the gKdV-k equations are only locally well-posed in when . In fact, the standard KdV equation is bi-hamiltonian.


If is even, the sign of is important. The case is known as the defocussing case, while is the focussing case. When is odd, the constant can always be scaled out, so we do not distinguish focussing and defocussing in this case.

Estimates

The perturbation theory for the KdV-type equations rests on a number of linear, bilinear, trilinear, or multilinear estimates for the Airy equation. These estimates involve a number of function space norms, such as the X^s,b spaces. See the page on Airy estimates for more details.