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==Equations of <span class="SpellE">Korteweg</span> de <span class="SpellE">Vries</span> type==
==Equations of Korteweg-de Vries type==


<div class="MsoNormal" style="text-align: center"><center>
The ''equations of Korteweg-de Vries type'' are all nonlinear perturbations of the [[Airy equation]].  They take the general form
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</center></div>


<center>'''Overview'''</center>
<center><math>\partial_t u + \partial_x^3 u + \partial_x P(u) = 0</math></center>


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. 


The <span class="SpellE">KdV</span> <span class="GramE">family of equations are</span> of the form
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.
<center><math>u_t + u_{xxx} + P(u)_x = 0</math></center>
<span class="GramE">where</span> u(<span class="SpellE">x,t</span>) is a function of one space and one time variable, and P(u) is some polynomial of u. 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. The function u and the polynomial P are usually assumed to be real.


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 [schrodinger.html#Cubic NLS on R 1D NLS-3] equation by making more limiting assumptions (in particular, weak nonlinearity and slowly varying amplitude).
== Specific equations ==


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


When <span class="GramE">P(</span>u) = c u^{k+1}, then the equation is referred to as generalized <span class="SpellE">gKdV</span> of order k, or <span class="SpellE">gKdV</span>-k. <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 class="SpellE">KdV</span> (<span class="SpellE">mKdV</span>) equation. <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>.
* 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]


If k is even, the sign of c is important. The c < 0 case is known as the <span class="SpellE">defocussing</span> case, while c > 0 is the <span class="SpellE">focussing</span> case. 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.
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>).


Drift terms <span class="SpellE">u_x</span> can be added, but they can be 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]. 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))).
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]].


<span class="SpellE">KdV</span>-type equations on R or T always come with three conserved quantities:
== History ==


<center><math>Mass:=\int u dx, L^2 size \int u^2 dx, Hamiltonian:=\int  u_x^2 - V(u) dx</math></center>
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).


<span class="GramE">where</span> V is a primitive of P. Note that the 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. The mass has no definite sign and so is only useful in specific cases (e.g. perturbations of a <span class="SpellE">soliton</span>).
== Conservation laws, symmetries, and criticality ==


In general, the above three quantities are the only conserved quantities available, but the [#kdv <span class="SpellE">KdV</span>] and [#mkdv <span class="SpellE">mKdV</span>] equations come with infinitely many more such conserved quantities due to their completely <span class="SpellE">integrable</span> nature.
KdV-type equations on R or T always come with three conserved quantities:


The critical (or scaling) regularity 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>


<center><span class="SpellE">s_c</span> = 1/2 - 2/k.</center>
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>).


In particular, [#kdv <span class="SpellE">KdV</span>], [#mkdv <span class="SpellE">mKdV</span>], 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. 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>). Note that H^1 is always a sub-critical regularity.
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 dispersion relation \<span class="SpellE">tau</span> = \xi^3 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 (L^2 will do). 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. 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 class="SpellE">solitons</span> as t -> infinity, but it is an interesting open question as to whether the same phenomenon occurs for the other <span class="SpellE">KdV</span>-type equations.
The [[critical]] (or scaling) regularity is


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
<center><math>s_c = \frac{1}{2} - \frac{2}{k}.</math></center>


<center>{<span class="GramE">u</span>, v} := \<span class="SpellE">int</span> u <span class="SpellE">v_x</span> <span class="SpellE">dx</span>.</center>
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.


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. 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.
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.


<div class="MsoNormal" style="text-align: center"><center>
== Symplectic structure ==
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</center></div>


<center>'''Airy estimates'''</center>
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
 
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
 
<center><tt><font size="10.0pt"><nowiki>|| u ||_{</nowiki><span class="SpellE">s<span class="GramE">,b</span></span>} = || <xi>^s <tau-xi^3>^b \hat{u} ||_2.</font></tt></center>
 
Linear space-time estimates in which the space norm is evaluated first are known as [#kdv_linear <span class="SpellE">Strichartz</span> estimates], but these estimates only play a minor role in the theory. A more important category of linear estimates are the smoothing estimates and maximal function estimates. The X^{<span class="SpellE">s<span class="GramE">,b</span></span>} spaces are used primarily for [#kdv_bilinear bilinear estimates], although more recently [#KdV_multilinear <span class="SpellE">multilinear</span> estimates have begun to appear]. These spaces and estimates first appear in the context of the <span class="SpellE">Schrodinger</span> equation in [[references.html#Bo1993b Bo1993b]], although the analogues spaces for the wave equation appeared earlier [[references.html#RaRe1982 RaRe1982]], [[references.html#Be1983 Be1983]] in the context of <span class="SpellE">propogation</span> of singularities. See also [[references.html#Bo1993 Bo1993]], [[references.html#KlMa1993 KlMa1993]].
 
<div class="MsoNormal" style="text-align: center"><center>
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<center>'''Linear Airy estimates'''</center>
 
* If u is in X^{0,1/2+} on '''R''', then
** u is in L^\<span class="SpellE">infty_t</span> L^2_x (energy estimate)
** <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>) [[references.html#KnPoVe1993 KnPoVe1993]]
** <span class="SpellE">D_x</span> u is in L^\<span class="SpellE">infty_x</span> L^2_t (sharp Kato smoothing effect) [[references.html#KnPoVe1993 KnPoVe1993]]. Earlier versions of this estimate were obtained in [[references.html#Ka1979b Ka1979b]], [[references.html#KrFa1983 KrFa1983]].
** <span class="SpellE">D_x</span>^{-1/4} u is in L^4_x L^\<span class="SpellE">infty_t</span> (Maximal function) [[references.html#KnPoVe1993 KnPoVe1993]], [[references.html#KnRu1983 KnRu1983]]
** <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) [[references.html#KnPoVe1993 KnPoVe1993]]
** ''Remark''<nowiki>: Further estimates are available by </nowiki><span class="SpellE">Sobolev</span>, differentiation, Holder, and interpolation. For instance:
*** <span class="SpellE">D_x</span> u is in L^2_{<span class="SpellE">x,t</span>} locally in space [[references.html#Ka1979b Ka1979b]] - use Kato and Holder (can also be proven directly by integration by parts)
*** u is in L^2_{<span class="SpellE">x,t</span>} locally in time - use energy and Holder
*** <span class="SpellE">D_x</span>^{3/4-} u is in L^8_x L^2_t locally in time - interpolate previous with Kato
*** <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)
*** <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>. (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>}).
*** u is in L^8_{<span class="SpellE">x,t</span>}- use previous and <span class="SpellE">Sobolev</span> in space
*** If u is in X^{0,1/3+}, then u is in L^4_{<span class="SpellE">x,t</span>} [[references.html#Bo1993b Bo1993b]] - interpolate previous with the trivial identity X^{0,0} = L^2
*** 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 [[references.html#Bo1993b Bo1993b]] - interpolate Kato with X^{0,0} = L^2
* If u is in X^{0,1/2+} on '''T''', then
** <span class="GramE">u</span> is in L^\<span class="SpellE">infty_t</span> L^2_x (energy estimate). This is also true in the large period case.
** <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) [[references.html#Bo1993b Bo1993b]].
** <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. [[references.html#Bo1993b Bo1993b]]. It is conjectured that this can be improved to L^8_{<span class="SpellE">x<span class="GramE">,t</span></span>}.
** ''Remark''<nowiki>: there is no smoothing on the circle, so one can never gain regularity.</nowiki>
* If u is in X^{0,1/2} on a circle with large period \lambda, then
** <span class="GramE">u</span> is in L^4_{<span class="SpellE">x,t</span>} locally in time, with a bound of \lambda^{0+}.
*** 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. [[references.html#CoKeStTaTk-p2 CoKeStTkTa-p2]]
 
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<center>'''Bilinear Airy estimates'''</center>
 
* The key algebraic fact is
 
<center><tt><font size="10.0pt">\xi_1^3 + \xi_2^3 + \xi_3^3 = 3 \xi_1 \xi_2 \xi_3</font></tt><br /><tt><font size="10.0pt">(whenever \xi_1 + \xi_2 + \xi_3 = 0)</font></tt></center>
 
* The -3/4+ estimate [[references.html#KnPoVe1996 KnPoVe1996]] on '''R'''<nowiki>:</nowiki>
 
<center><tt><font size="10.0pt"><nowiki>|| (</nowiki><span class="SpellE">uv</span><span class="GramE">)_</span>x ||_{-3/4+, -1/2+} <~ || u ||_{-3/4+, 1/2+} || v ||_{-3/4+, 1/2+}</font></tt></center>
 
<math>\| u v \|_{-3/4+, -1/2+} <~ \| u \|_{-3/4+, 1/2+} \| v \|_{-3/4+, 1/2+}</math>
 
** The above estimate fails at the endpoint -3/4. [[references.html#NaTkTs-p NaTkTs2001]]
** As a corollary of this estimate we have the -3/8+ estimate [[references.html#CoStTk1999 CoStTk1999]] on '''R'''<nowiki>: If u and v have no low frequencies ( |\xi| <~ 1 ) then</nowiki>
 
<center><tt><font size="10.0pt"><nowiki>|| (</nowiki><span class="SpellE">uv</span><span class="GramE">)_</span>x ||_{0, -1/2+} <~ || u ||_{-3/8+, 1/2+} || v ||_{-3/8+, 1/2+}</font></tt></center>
 
* The -1/2 estimate [[references.html#KnPoVe1996 KnPoVe1996]] on '''T'''<nowiki>: if </nowiki><span class="SpellE">u,v</span> have mean zero, then for all s >= -1/2
 
<center><tt><font size="10.0pt"><nowiki>|| (</nowiki><span class="SpellE">uv</span><span class="GramE">)_</span>x ||_{s, -1/2} <~ || u ||_{s, 1/2} || v ||_{s, 1/2}</font></tt></center>
 
** The above estimate fails for s < -1/2. Also, one cannot replace 1/2, -1/2 by 1/2+, -1/2+. [[references.html#KnPoVe1996 KnPoVe1996]]
** This estimate also holds in the large period case if one is willing to lose a power of \lambda^{0+} in the constant. [[references.html#CoKeStTaTk-p2 CoKeStTkTa-p2]]
* ''Remark''<nowiki>: In principle, a complete list of bilinear estimates could be obtained from [</nowiki>[references.html#Ta-p2 Ta-p2]].
 
<div class="MsoNormal" style="text-align: center"><center>
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<center><span class="SpellE">'''Trilinear'''</span>''' Airy estimates'''</center>
 
* The key algebraic fact is (various permutations of)
 
<center><tt><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)</font></tt><br /><tt><font size="10.0pt">(whenever \xi_1 + \xi_2 + \xi_3 + \xi_4 = 0)</font></tt></center>
 
* The 1/4 estimate [[references.html#Ta-p2 Ta-p2]] on '''R'''<nowiki>:</nowiki>
 
<center><tt><font size="10.0pt"><nowiki>|| (</nowiki><span class="SpellE">uvw</span><span class="GramE">)_</span>x ||_{1/4, -1/2+} <~ || u ||_{1/4, 1/2+} || v ||_{1/4, 1/2+} || w ||_{1/4, 1/2+}</font></tt></center>
 
The 1/4 is sharp [[references.html#KnPoVe1996 KnPoVe1996]].We also have
 
<center><tt><font size="10.0pt"><nowiki>|| </nowiki><span class="SpellE"><span class="GramE">uv<u>w</u></span></span> ||_{-1/4, -5/12+} <~ || u ||_{-1/4, 7/12+} || v ||_{-1/4, 7/12+} || w ||_{-1/4, 7/12+}</font></tt></center>
 
<span class="GramE">see</span> [<span class="SpellE">Cv</span>-p].
 
* The 1/2 estimate [[references.html#CoKeStTaTk-p3 CoKeStTkTa-p3]] on '''T'''<nowiki>: if </nowiki><span class="SpellE">u,v,w</span> have mean zero, then
 
<center><tt><font size="10.0pt"><nowiki>|| (</nowiki><span class="SpellE">uvw</span><span class="GramE">)_</span>x ||_{1/2, -1/2} <~ || u ||_{1/2, 1/2*} || v ||_{1/2, 1/2*} || w ||_{1/2, 1/2*}</font></tt></center>
 
The 1/2 is sharp [[references.html#KnPoVe1996 KnPoVe1996]].
 
* ''Remark''<nowiki>: the </nowiki><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>.
 
<div class="MsoNormal" style="text-align: center"><center>
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</center></div>
 
<center><span class="SpellE">'''Multilinear'''</span>''' Airy estimates'''</center>
 
* We have the <span class="SpellE">quintilinear</span> estimate on '''R'''<nowiki>: [</nowiki>[references.html#CoKeStTaTk-p2 CoKeStTkTa-p2]]
 
<center><tt><font size="10.0pt">\<span class="SpellE">int</span> u^3 v^2 <span class="SpellE">dx</span> <span class="SpellE">dt</span> <~ || u ||<span class="GramE">_{</span>1/4+,1/2+}^3 || v ||_{-3/4+,1/2+}^2</font></tt></center>
 
* The analogue for this on '''T''' is: [[references.html#CoKeStTaTk-p2 CoKeStTkTa-p2], [references.html#CoKeStTaTk-p3 CoKeStTkTa-p3]]
 
<center><tt><font size="10.0pt">\<span class="SpellE">int</span> u^3 v^2 <span class="SpellE">dx</span> <span class="SpellE">dt</span> <~ || u ||<span class="GramE">_{</span>1/2,1/2*}^3 || v ||_{-1/2,1/2*}^2</font></tt></center>
 
In fact, this estimate also holds for large period, but a loss of lambda<span class="GramE">^{</span>0+}.
 
<div class="MsoNormal" style="text-align: center"><center>
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</center></div>
 
<center>'''The <span class="SpellE">KdV</span> equation'''</center>
 
The <span class="SpellE">KdV</span> equation is
 
<center><span class="SpellE">u_t</span> + <span class="SpellE">u_xxx</span> + u <span class="SpellE">u_x</span> = 0.</center>
 
It is completely <span class="SpellE">integrable</span>, and has infinitely many conserved quantities. 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.
 
The <span class="SpellE">KdV</span> equation has been studied on the [#kdv_on_R line], the [#kdv_on_T circle], and the [#KdV_on_R+ half-line].
 
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<center><span class="SpellE">'''KdV'''</span>''' on R'''</center>
 
* Scaling is <span class="SpellE">s_c</span> = -3/2.
* LWP in <span class="SpellE">H^s</span> for s >= -3/4 [<span class="SpellE">CtCoTa</span>-p], using a modified Miura transform and the [#mKdV_on_R <span class="SpellE">mKdV</span> theory]. This is despite the failure of the key bilinear estimate [[references.html#NaTkTs-p NaTkTs2001]]
** For s within a logarithm for s=-3/4 [<span class="SpellE">MurTao</span>-p].
** Was proven for s > -3/4 [[references.html#KnPoVe1996 KnPoVe1996]].
** Was proven for s > -5/8 in [[references.html#KnPoVe1993b KnPoVe1993b]].
** Was proven for s >= 0 in [[references.html#Bo1993b Bo1993b]].
** Was proven for s > 3/4 in [[references.html#KnPoVe1993 KnPoVe1993]].
** Was proven for s > 3/2 in [[references.html#BnSm1975 BnSm1975]], [[references.html#Ka1975 Ka1975]], [[references.html#Ka1979 Ka1979]], [[references.html#GiTs1989 GiTs1989]], [[references.html#Bu1980 Bu1980]]<span class="GramE">, ....</span>
** 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 < -3/4 (in the complex setting) by <span class="SpellE">soliton</span> examples [[references.html#KnPoVe-p <span class="SpellE">KnPoVe</span>-p]].
*** 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 [#mKdV_on_R corresponding result for <span class="SpellE">mKdV</span>].
*** Below -3/4 the solution map was known to not be C^3 [[references.html#Bo1993b Bo1993b]], [[references.html#Bo1997 Bo1997]]; this was refined to C^2 in [[references.html#Tz1999b Tz1999b]].
** When the initial data is a real, rapidly decreasing measure one has a global smooth solution for t > 0 [[referencs.html#Kp1993 Kp1993]]. Without the rapidly decreasing hypothesis one can still construct a global weak solution [[references.html#Ts1989 Ts1989]]
* GWP in <span class="SpellE">H^s</span> for s > -3/4 (if u is real) [[references.html#CoKeStTaTk2003 CoKeStTkTa2003]].
** Was proven for s > -3/10 in [[references.html#CoKeStTkTa2001 CoKeStTkTa2001]]
** Was proven for s>= 0 in [[references.html#Bo1993b Bo1993b]]. Global weak solutions in L^2 were constructed in [[references.html#Ka1983 Ka1983]], [[references.html#KrFa1983 KrFa1983]], and were shown to obey the expected local smoothing estimate. These weak solutions were shown to be unique in [[references.html#Zh1997b Zh1997b]]
** Was proven for s>= 1 in [[references.html#KnPoVe1993 KnPoVe1993]].
** Was proven for s>= 2 in [[references.html#BnSm1975 BnSm1975]], [[references.html#Ka1975 Ka1975]], [[references.html#Ka1979 Ka1979]]<span class="GramE">, ....</span>
** ''Remark''<nowiki>: In the complex setting GWP fails for large data with Fourier support on the half-line [Bona/</nowiki><span class="SpellE">Winther</span>?], [<span class="SpellE">Birnir</span>]<span class="GramE">, ????</span>. This result extends to a wide class of dispersive PDE.
* 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.
* <span class="SpellE">Solitons</span> are <span class="SpellE">orbitally</span> H^1 stable [[references.html#Bj1972 Bj1972]]
** In <span class="SpellE">H^s</span>, 0 <= s < 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]
** In L^2, orbital stability has been obtained in [[references.html#MeVe2003 MeVe2003]].
 
The <span class="SpellE">KdV</span> equation can also be generalized to a 2x2 system
 
<center><span class="SpellE"><tt><font size="10.0pt">u_t</font></tt></span><tt><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</font></tt><br /><tt><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></font></tt></center>
 
<span class="GramE">where</span> b_1,b_2 are positive constants and a_1,a_2,a_3,r are real constants. This system was introduced in [[references.html#GeaGr1984 GeaGr1984]] to study strongly interacting pairs of weakly nonlinear long waves, and studied further in [[references.html#BnPoSauTm1992 BnPoSauTm1992]]. In [[references.html#AsCoeWgg1996 AsCoeWgg1996]] 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.
 
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<center><span class="SpellE">'''KdV'''</span>''' on R^+'''</center>
 
* The <span class="SpellE">KdV</span> Cauchy-boundary problem on the half-line is
 
<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; u(x,0) = u_0(x); u(0,t) = h(t)</center>
 
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. The drift term <span class="SpellE">u_x</span> appears naturally from the derivation of <span class="SpellE">KdV</span> from fluid mechanics. (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).
 
* Because one is restricted to the half-line, it becomes a little tricky to use the Fourier transform. One approach is to use the Fourier-<span class="SpellE">Laplace</span> transform instead.
* Some compatibility conditions between u_0 and h are needed. The higher the regularity, the more compatibility conditions are needed. 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}. (Remember that time has dimensions ''length''^3).
* 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 >= 0 [<span class="SpellE">CoKe</span>-p], assuming compatibility. The drift term may be omitted because of the time localization.
** For s > 3/4 this was proven in [[references.html#BnSuZh-p <span class="SpellE">BnSuZh</span>-p]] (assuming that there is a drift term).
** Was proven for data in sufficiently weighted H^1 spaces in [[references.html#Fa1983 Fa1983]].
** From the real line theory one might expect to lower this to -3/4, but there appear to be technical difficulties with this.
* GWP is known for initial data in L^2 and boundary data in H<span class="GramE">^{</span>7/12}, assuming compatibility.
** for initial data in H^1 and boundary data in H^{5/6}_loc this was proven in [[references.html#BnSuZh-p <span class="SpellE">BnSuZh</span>-p]]
** Was proven for smooth data in [[references.html#BnWi1983 BnWi1983]]
 
<div class="MsoNormal" style="text-align: center"><center>
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</center></div>
 
<center><span class="SpellE">'''KdV'''</span>''' on T'''</center>
 
* Scaling is <span class="SpellE">s_c</span> = -3/2.
* C^0 LWP in <span class="SpellE">H^s</span> for s >= -1, assuming u is real [<span class="SpellE">KpTp</span>-p]
** C^0 LWP in <span class="SpellE">H^s</span> for s >= -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>?]
** Analytic LWP in <span class="SpellE">H^s</span> for s >= -1/2, in the complex case [[references.html#KnPoVe1996 KnPoVe1996]]. 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>0.
** Analytic LWP was proven for s >= 0 in [[references.html#Bo1993b Bo1993b]].
** Analytic ill <span class="SpellE">posedness</span> at s<-1/2, even in the real case [[references.html#Bo1997 Bo1997]]
*** This has been refined to failure of uniform continuity at s<-1/2 [<span class="SpellE">CtCoTa</span>-p]
** 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. In particular one has <span class="SpellE">symplectic</span> non-squeezing [CoKeStTkTa-p9], [[references.html#Bo1999 Bo1999]].
* C^0 GWP in <span class="SpellE">H^s</span> for s >= -1, in the real case [<span class="SpellE">KpTp</span>-p].
** Analytic GWP in <span class="SpellE">H^s</span> in the real case for s >= -1/2 [[references.html#CoKeStTaTk-p2 CoKeStTkTa-p2]]; see also [[references.html#CoKeStTaTk-p3 CoKeStTkTa-p3]].
** A short proof for the s > -3/10 case is in [[references.html#CoKeStTaTk-p2a CoKeStTkTa-p2a]]
** Was proven for s >= 0 in [[references.html#Bo1993b Bo1993b]].
** GWP for real initial data which are measures of small norm [[references.html#Bo1997 Bo1997]] <span class="GramE">The</span> small norm restriction is presumably technical.
*** ''Remark''<nowiki>: measures have the same scaling as H</nowiki><span class="GramE">^{</span>-1/2}, but neither space includes the other. (Measures are in H<span class="GramE">^{</span>-1/2-\eps} though).
** One has GWP for real random data whose Fourier coefficients decay like 1/|k| (times a Gaussian random variable) [[references.html#Bo1995c Bo1995c]]. Indeed one has an invariant measure.
** <span class="SpellE">Solitons</span> are asymptotically H^1 stable [MtMe-p3], [<span class="SpellE">MtMe</span>-p].Indeed, the solution decouples into a finite sum of <span class="SpellE">solitons</span> plus dispersive radiation [[references.html#EckShr1988 EckShr1988]]
 
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<center>'''The modified <span class="SpellE">KdV</span> equation'''</center>
 
The (<span class="SpellE">defocussing</span>) <span class="SpellE">mKdV</span> equation is
 
<center><span class="SpellE">u_t</span> + <span class="SpellE">u_xxx</span> = 6 u^2 <span class="SpellE">u_x</span>.</center>
 
It is completely <span class="SpellE">integrable</span>, and has infinitely many conserved quantities. 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. This equation has been studied on the [#mKdV_on_R line], [#mKdV_on_T circle], and [#gKdV_on_R+ half-line].
 
The ''Miura transformation'' 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 [#kdv <span class="SpellE">KdV</span>]
 
<center><span class="SpellE">v_t</span> + <span class="SpellE">v_xxx</span> = 6 v <span class="SpellE">v_x</span>.</center>
 
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>.
 
The <span class="SpellE">focussing</span> <span class="SpellE">mKdV</span>
 
<center><span class="SpellE">u_t</span> + <span class="SpellE">u_xxx</span> = - 6 u^2 <span class="SpellE">u_x</span></center>
 
<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. This transforms <span class="SpellE">focussing</span> <span class="SpellE">mKdV</span> to ''complex-valued'' <span class="SpellE">KdV</span>, which is a slightly less tractable equation. (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).
 
The Miura transformation can be generalized. If v and w solve the system
 
<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></center>
 
Then u = v^2 + <span class="SpellE">v_x</span> + w is a solution of <span class="SpellE">KdV</span>. In particular, if a and b are constants and v solves
 
<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></center>
 
<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 ''Gardener transform'').
 
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<center><span class="SpellE"><span class="GramE">'''mKdV'''</span></span>''' on R and R^+'''</center>
 
* Scaling is <span class="SpellE">s_c</span> = -1/2.
* LWP in <span class="SpellE">H^s</span> for s >= 1/4 [[references.html#KnPoVe1993 KnPoVe1993]]
** Was shown for s>3/2 in [[references.html#GiTs1989 GiTs1989]]
** This is sharp in the <span class="SpellE">focussing</span> case [[references.html#KnPoVe-p <span class="SpellE">KnPoVe</span>-p]], in the sense that the solution map is no longer uniformly continuous for s < 1/4.
*** 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 [schrodinger.html#Cubic NLS on R NLS]. (This high frequency approximation has also been utilized in [[references.html#Sch1998 Sch1998]]).
*** Below 1/4 the solution map was known to not be C^3 in [[references.html#Bo1993b Bo1993b]], [[references.html#Bo1997 Bo1997]].
** 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.
** Global weak solutions in L^2 were constructed in [[references.html#Ka1983 Ka1983]]. Thus in L^2 one has global existence but no uniform continuity.
** Uniqueness is also known when the initial data lies in the weighted space <x>^{3/8} u_0 in L^2 [[references.html#GiTs1989 GiTs1989]]
** LWP has also been demonstrated when <xi>^s hat(u_0) lies in L^{r’} for 4/3 < r <= 2 and s >= ½ - 1/2r [Gr-p4]
* GWP in <span class="SpellE">H^s</span> for s > 1/4 [[references.html#CoKeStTaTk-p2 CoKeStTkTa-p2]], 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.
** Was proven for s>3/5 in [[references.html#FoLiPo1999 FoLiPo1999]]
** Is implicit for s >= 1 from [[references.html#KnPoVe1993 KnPoVe1993]]
** On the half-line GWP is known when s >= 1 and the boundary data is in H^{11/12}, assuming compatibility and small L^2 norm [<span class="SpellE">CoKe</span>-p]
** 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].
** <span class="SpellE">Solitions</span> are asymptotically H^1 stable [MtMe-p3], [<span class="SpellE">MtMe</span>-p]
 
<div class="MsoNormal" style="text-align: center"><center>
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</center></div>
 
<center><span class="SpellE"><span class="GramE">'''mKdV'''</span></span>''' on T'''</center>
 
* Scaling is <span class="SpellE">s_c</span> = -1/2.
* C^0 LWP in L^2 in the defocusing case [KpTp-p2]
** C^0 LWP in <span class="SpellE">H^s</span> for s > 3/8 [Takaoka and <span class="SpellE">Tsutsumi</span>?]Note one has to gauge away a nonlinear resonance term before one can apply iteration methods.
** Analytic LWP in <span class="SpellE">H^s</span> for s >= 1/2, in both focusing and defocusing cases [[references.html#KnPoVe1993 KnPoVe1993]], [[references.html#Bo1993b Bo1993b]].
** This is sharp in the sense of analytic well-<span class="SpellE">posedness</span> [[references.html#KnPoVe1996 KnPoVe1996]] or uniform well-<span class="SpellE">posedness</span> [<span class="SpellE">CtCoTa</span>-p]
* C^0 GWP in L^2 in the defocusing case [KpTp-p2]
** Analytic GWP in <span class="SpellE">H^s</span> for s >= 1/2<span class="GramE"> [</span>[references.html#CoKeStTaTk-p2 CoKeStTkTa-p2]], 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.
** Was proven for s >= 1 in [[references.html#KnPoVe1993 KnPoVe1993]], [[references.html#Bo1993b Bo1993b]].
** One has GWP for random data whose Fourier coefficients decay like 1/|k| (times a Gaussian random variable) [[references.html#Bo1995c Bo1995c]]. Indeed one has an invariant measure. 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).
 
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<center><span class="GramE">'''gKdV_3'''</span>''' on R and R^+'''</center>
 
* Scaling is <span class="SpellE">s_c</span> = -1/6.
* LWP in <span class="SpellE">H^s</span> for s > -1/6 [Gr-p3]
** Was shown for s>=1/12 [[references.html#KnPoVe1993 KnPoVe1993]]
** Was shown for s>3/2 in [[references.html#GiTs1989 GiTs1989]]
** The result s >= 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..
* GWP in <span class="SpellE">H^s</span> for s >= 0 [Gr-p3]
** For s>=1 this is in [[references.html#KnPoVe1993 KnPoVe1993]]
** Presumably one can use either the Fourier truncation method or the "I-method" to go below L^2. Even though the equation is not completely <span class="SpellE">integrable</span>, the one-dimensional nature of the equation suggests that "correction term" techniques will also be quite effective.
** On the half-line GWP is known when s >= 1 and the boundary data is in H^{5/4}, assuming compatibility and small L^2 norm [<span class="SpellE">CoKe</span>-p]
* <span class="SpellE">Solitons</span> are H^1-stable [[references.html#CaLo1982 CaLo1982]], [[references.html#Ws1986 Ws1986]], [[references.html#BnSouSr1987 BnSouSr1987]] and asymptotically H^1 stable [MtMe-p3], [<span class="SpellE">MtMe</span>-p]
 
<div class="MsoNormal" style="text-align: center"><center>
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<center><span class="GramE">'''gKdV_3'''</span>''' on T'''</center>
 
* Scaling is <span class="SpellE">s_c</span> = -1/6.
* LWP in <span class="SpellE">H^s</span> for s>=1/2 [[references.html#CoKeStTaTk-p3 CoKeStTkTa-p3]]
** Was shown for s >= 1 in [[references.html#St1997c St1997c]]
** One has analytic ill-<span class="SpellE">posedness</span> for s<1/2 [[references.html#CoKeStTaTk-p3 CoKeStTkTa-p3]] by a modification of the example in [[references.html#KnPoVe1996 KnPoVe1996]].
* GWP in <span class="SpellE">H^s</span> for s>5/6 [[references.html#CoKeStTaTk-p3 CoKeStTkTa-p3]]
** Was shown for s >= 1 in [[references.html#St1997c St1997c]]
** This result may well be improvable by the "damping correction term" method in<span class="GramE"> [</span>[references.html#CoKeStTaTk-p2 CoKeStTkTa-p2]].
* ''Remark''<nowiki>: For this equation it is convenient to make a "gauge transformation'' to subtract off the mean of </nowiki><span class="GramE">P(</span>u).
 
<div class="MsoNormal" style="text-align: center"><center>
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</center></div>
 
<center><span class="GramE">'''gKdV_4'''</span>''' on R and R^+'''</center>
 
(Thanks to Felipe <span class="SpellE">Linares</span> for help with the references here - Ed.)A good survey for the results here is in [Tz-p2].
 
* Scaling is <span class="SpellE">s_c</span> = 0 (i.e. L^2-critical).
* LWP in <span class="SpellE">H^s</span> for s >= 0 [[references.html#KnPoVe1993 KnPoVe1993]]
** Was shown for s>3/2 in [[references.html#GiTs1989 GiTs1989]]
** The same result s >= 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..
* GWP in <span class="SpellE">H^s</span> for s > 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].
** For s >= 1 and the defocusing case this is in [[references.html#KnPoVe1993 KnPoVe1993]]
** 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]. In such a case the blowup profile must approach the ground state (modulo <span class="SpellE">scalings</span> and translations), see [MtMe-p4], [[references.html#MtMe2001 MtMe2001]]. 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.
** Explicit self-similar blow-up solutions have been constructed [<span class="SpellE">BnWe</span>-p] but these are not in L^2.
** GWP for small L^2 data in either case [[references.html#KnPoVe1993 KnPoVe1993]]. 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). It seems like a reasonable (but difficult) conjecture to have GWP for large L^2 data in the defocusing case.
** On the half-line GWP is known when s >= 1 and the boundary data is in H^{11/12}, assuming compatibility and small L^2 norm [<span class="SpellE">CoKe</span>-p]
* <span class="SpellE">Solitons</span> are H^1-unstable [[references.html#MtMe2001 MtMe2001]]. 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 [[references.html#MtMe-p <span class="SpellE">MtMe</span>-p]].
 
<div class="MsoNormal" style="text-align: center"><center>
----
</center></div>
 
<center><span class="GramE">'''gKdV_4'''</span>''' on T'''</center>
 
* Scaling is <span class="SpellE">s_c</span> = 0.
* LWP in <span class="SpellE">H^s</span> for s>=1/2 [[references.html#CoKeStTaTk-p3 CoKeStTkTa-p3]]
** Was shown for s >= 1 in [[references.html#St1997c St1997c]]
** Analytic well-<span class="SpellE">posedness</span> fails for s < 1/2; this is essentially in [[references.html#KnPoVe1996 KnPoVe1996]]
* GWP in <span class="SpellE">H^s</span> for s>=1 [[references.html#St1997c St1997c]]
** This is almost certainly improvable by the techniques in [[references.html#CoKeStTaTk-p3 CoKeStTkTa-p3]], probably to s > 6/7. There are some low-frequency issues which may require the techniques in [[references.html#KeTa-p <span class="SpellE">KeTa</span>-p]].
* ''Remark''<nowiki>: For this equation it is convenient to make a "gauge transformation'' to subtract off the mean of </nowiki><span class="GramE">P(</span>u).
 
<div class="MsoNormal" style="text-align: center"><center>
----
</center></div>
 
<center><span class="SpellE"><span class="GramE">'''gKdV'''</span></span>''' on R^+'''</center>
 
* The <span class="SpellE">gKdV</span> Cauchy-boundary problem on the half-line is
 
<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; u(x,0) = u_0(x); u(0,t) = h(t)</center>
 
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. The drift term <span class="SpellE">u_x</span> is convenient for technical reasons; it is not known whether it is truly necessary.
 
* 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 > 3/4 [<span class="SpellE">CoKn</span>-p].
** The techniques are based on [[references.html#KnPoVe1993 KnPoVe1993]] and a replacement of the IVBP with a forced IVP.
** This has been improved to s >= <span class="SpellE">s_c</span> = 1/2 - 2/k when k > 4 [<span class="SpellE">CoKe</span>-p].
** For [#KdV_on_R+ <span class="SpellE">KdV</span>], [#mKdV_on_R <span class="SpellE">mKdV</span>], [#gKdV_3_on_R gKdV-3] , and [#gKdV_4_on_R gKdV-4] see the corresponding sections on this page.
 
<div class="MsoNormal" style="text-align: center"><center>
----
</center></div>
 
<center>'''Nonlinear <span class="SpellE">Schrodinger</span>-Airy equation'''</center>
 
The equation
 
<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></center>
 
<span class="GramE">on</span> R is a combination of the [schrodinger.html#Cubic_NLS_on_R cubic NLS equation] , the [schrodinger.html#dnls-3_on_R derivative cubic NLS equation], [#mKdV_on_R complex <span class="SpellE">mKdV</span>], and a cubic nonlinear Airy equation.This equation is a general model for <span class="SpellE">propogation</span> of pulses in an optical fiber [[references.html#Kod1985 Kod1985]], [[references.html#HasKod1987 HasKod1987]]
 
<span style="mso-fareast-font-family: Symbol; mso-bidi-font-family: Symbol"><font face="Symbol"><span style="mso-list: Ignore">·</span></font></span>When c=delta=epsilon = 0, scaling is s=-1.When c=gamma=0, scaling is –1/2.
 
<span style="mso-fareast-font-family: Symbol; mso-bidi-font-family: Symbol"><font face="Symbol"><span style="mso-list: Ignore">·</span></font></span>LWP is known when s >= ¼ [[references.html#St1997d St1997d]]
 
<span style="mso-fareast-font-family: &quot;Courier New&quot;"><font face="&quot;Courier New&quot;"><span style="mso-list: Ignore">o</span></font></span>For s > ¾ this is in [[references.html#Lau1997 Lau1997]], [[references.html#Lau2001 Lau2001]]
 
<span style="mso-fareast-font-family: &quot;Courier New&quot;"><font face="&quot;Courier New&quot;"><span style="mso-list: Ignore">o</span></font></span>The s>=1/4 result is also known when c is a time-dependent function [Cv2002], [CvLi2003]
 
<span style="mso-fareast-font-family: &quot;Courier New&quot;"><font face="&quot;Courier New&quot;"><span style="mso-list: Ignore">o</span></font></span>For s < -1/4 and delta or epsilon non-zero, the solution map is not C^3 [<span class="SpellE">CvLi</span>-p]
 
<span style="mso-fareast-font-family: &quot;Courier New&quot;"><font face="&quot;Courier New&quot;"><span style="mso-list: Ignore">o</span></font></span>When delta = epsilon = 0 LWP is known for s > -1/4 [[references.html#Cv2004 Cv2004]]
 
<span style="mso-fareast-font-family: Wingdings; mso-bidi-font-family: Wingdings"><font face="Wingdings"><span style="mso-list: Ignore">§</span></font></span>For s < -1/4 the solution map is not C^3 [<span class="SpellE">CvLi</span>-p]
 
<div class="MsoNormal" style="text-align: center"><center>
----
</center></div>
 
<center>'''Miscellaneous <span class="SpellE">gKdV</span> results'''</center>
 
[Thanks to <span class="SpellE">Nikolaos</span> <span class="SpellE">Tzirakis</span> for some corrections - Ed.]
 
* On R with k > 4, <span class="SpellE">gKdV</span>-k is LWP down to scaling: s >= <span class="SpellE">s_c</span> = 1/2 - 2/k [[references.html#KnPoVe1993 KnPoVe1993]]
** Was shown for s>3/2 in [[references.html#GiTs1989 GiTs1989]]
** One has ill-<span class="SpellE">posedness</span> in the supercritical regime [[references.html#BirKnPoSvVe1996 BirKnPoSvVe1996]]
** For small data one has scattering [[references.html#KnPoVe1993c KnPoVe1993c]].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.
** <span class="SpellE">Solitons</span> are H^1-unstable [[references.html#BnSouSr1987 BnSouSr1987]]
** 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 > 1/2 [[references.html#St1995 St1995]]
* On R with any k, <span class="SpellE">gKdV</span>-k is GWP in <span class="SpellE">H^s</span> for s >= 1 [[references.html#KnPoVe1993 KnPoVe1993]], though for k >= 4 one needs the L^2 norm to be small; global weak solutions were constructed much earlier, with the same smallness assumption when k >= 4. This should be improvable below H^1 for all k.
* 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 >= 1 [[references.html#St1997b St1997b]]
* 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 [[references.html#KnPoVe-p3 KnPoVe-p3]], [KnPoVe-p4].
** In the completely <span class="SpellE">integrable</span> cases k=1,2 this is in [[references.html#Zg1992 Zg1992]]
** Also, a non-zero solution to <span class="SpellE">gKdV</span> cannot vanish on a rectangle in <span class="SpellE">spacetime</span> [[references.html#SauSc1987 SauSc1987]]; see also [[references.html#Bo1997b Bo1997b]].
** Extensions to higher order <span class="SpellE">gKdV</span> type equations are in [[references.html#Bo1997b Bo1997b]], [KnPoVe-p5].
* 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 > (19 - <span class="SpellE">sqrt</span>(57))/4 ~ 2.8625... [[references.html#CtWs1991 CtWs1991]]
** A similar result for k > (5+<span class="GramE">sqrt(</span>73))/4 ~ 3.39... <span class="GramE">was</span> obtained in [[references.html#PoVe1990 PoVe1990]].
** 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 [[references.html#AbSe1977 AbSe1977]]
* In the L^2 <span class="SpellE">subcritical</span> case 0 < k < 4, <span class="SpellE">multisoliton</span> solutions are asymptotically H^1-stable [<span class="SpellE">MtMeTsa</span>-p]
** For a single <span class="SpellE">soliton</span> this is in [MtMe-p3], [<span class="SpellE">MtMe</span>-p], [[references.html#Miz2001 Miz2001]]; earlier work is in [[references.html#Bj1972 Bj1972]], [[references.html#Bn1975 Bn1975]], [[references.html#Ws1986 Ws1986]], [[references.html#PgWs1994 PgWs1994]]
* A dissipative version of <span class="SpellE">gKdV</span>-k was analyzed in [[references.html#MlRi2001 MlRi2001]]
 
* 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 >= 1 [[references.html#St1997b St1997b]]
* On T with k >= 3, <span class="SpellE">gKdV</span>-k is LWP for s >= 1/2 [[references.html#CoKeStTaTk-p3 CoKeStTkTa-p3]]
** Was shown for s >= 1 in [[references.html#St1997c St1997c]]
** Analytic well-<span class="SpellE">posedness</span> fails for s < 1/2 [[references.html#CoKeStTaTk-p3 CoKeStTkTa-p3]], [[references.html#KnPoVe1996 KnPoVe1996]]
** For arbitrary smooth non-<span class="SpellE">linearities</span>, weak H^1 solutions were constructed in [[references.html#Bo1993b Bo1993]].
* On T with k >= 3, <span class="SpellE">gKdV</span>-k is GWP for s >= 1 except in the <span class="SpellE">focussing</span> case [[references.html#St1997c St1997c]]
** The estimates in [[references.html#CoKeStTaTk-p3 CoKeStTkTa-p3]] suggest that this is improvable to 13/14 - 2/7k, but this has only been proven in the sub-critical case k=3 [[references.html#CoKeStTaTk-p3 CoKeStTkTa-p3]]. In the critical and super-critical cases there are some low-frequency issues which may require the techniques in [[references.html#KeTa-p <span class="SpellE">KeTa</span>-p]].
 
<div class="MsoNormal" style="text-align: center"><center>
----
</center></div>
 
<center>'''The <span class="SpellE">KdV</span> Hierarchy'''</center>
 
The <span class="SpellE">KdV</span> equation
 
<center><span class="SpellE">V_t</span> + <span class="SpellE">V_xxx</span> = 6 <span class="SpellE">V_x</span></center>
 
<span class="GramE">can</span> be rewritten in the Lax Pair form
 
<center><span class="SpellE">L_t</span> = [L, P]</center>
 
<span class="GramE">where</span> L is the second-order operator
 
<center>L = -D^2 + V</center>
 
(D = d/<span class="SpellE">dx</span>) and P is the third-order <span class="SpellE">antiselfadjoint</span> operator
 
<center>P = 4D^3 + 3(DV + VD).</center>
 
(<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}).
 
One can replace P with other fractional powers of L. For instance, the <span class="SpellE">zeroth</span> order and higher terms of 4i L<span class="GramE">^{</span>5/2} are
 
<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)</center>
 
<span class="GramE">and</span> the Lax pair equation becomes
 
<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</center>
 
<span class="GramE">with</span> Hamiltonian
 
<center><span class="GramE">H(</span>V) = \<span class="SpellE">int</span> V_xx^2 - 5 V^2 <span class="SpellE">V_xx</span> - 5 V^4.</center>
 
These flows all commute with each <span class="GramE">other,</span> and their Hamiltonians are conserved by all the flows simultaneously.
 
The <span class="SpellE">KdV</span> <span class="GramE">hierarchy are</span> examples of higher order water wave models; a general formulation is
 
<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)</center>
 
<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.LWP for these equations in high regularity <span class="SpellE">Sobolev</span> spaces is in [[references.html#KnPoVe1994 KnPoVe1994]], and independently by <span class="SpellE"><span class="GramE">Cai</span></span> (ref?); see also [[references.html#CrKpSr1992 CrKpSr1992]].The case j=2 was studied by <span class="SpellE">Choi</span> (ref?).The non-scalar diagonal case was treated in [[references.html#KnSt1997 KnSt1997]]; the periodic case was studied in [Bo-p3].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]
 
<div class="MsoNormal" style="text-align: center"><center>
----
</center></div>
 
<center>'''Benjamin-Ono equation'''</center>
 
[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]
 
The ''generalized Benjamin-Ono equation'' <span class="SpellE">BO_a</span> is the scalar equation
 
<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</center>
 
<span class="GramE">where</span> <span class="SpellE">D_x</span> = <span class="SpellE">sqrt</span>{-Delta} is the positive differentiation operator. When a=1 this is [#kdv <span class="SpellE">KdV</span>]<nowiki>; when a=0 this is the Benjamin-Ono equation (BO) [</nowiki>[references.html#Bj1967 Bj1967]], [[references.html#On1975 On1975]], which models one-dimensional internal waves in deep water. Both of these equations are completely <span class="SpellE">integrable</span> (see e.g. [[references.html#AbFs1983 AbFs1983]], [[references.html#CoiWic1990 CoiWic1990]]), though the intermediate cases 0 < a < 1 are not.
 
When a=0, scaling is s = -1/2, and the following results are known:


* LWP in <span class="SpellE">H^s</span> for s >= 1 [Ta-p]
<center><math>(u,v) := \int u \partial_x^{-1} v dx</math>.</center>
** For s >= 9/8 this is in [<span class="SpellE">KnKoe</span>-p]
** For s >= 5/4 this is in [<span class="SpellE">KocTz</span>-p]
** For s >= 3/2 this is in [[references.html#Po1991 Po1991]]
** For s > 3/2 this is in [[references.html#Io1986 Io1986]]
** For s > 3 this is in [[references.html#Sau1979 Sau1979]]
** For no value of s is the solution map uniformly continuous [KocTz-p2]
*** For s < -1/2 this is in [<span class="SpellE">BiLi</span>-p]
* Global weak solutions exist for L^2 data [[references.html#Sau1979 Sau1979]], [[references.html#GiVl1989b GiVl1989b]], [[references.html#GiVl1991 GiVl1991]], [[references.html#Tom1990 Tom1990]]
* Global well-<span class="SpellE">posedness</span> in <span class="SpellE">H^s</span> for s >= 1 [Ta-p]
** For s >= 3/2 this is in [[references.html#Po1991 Po1991]]
** For smooth solutions this is in [[references.html#Sau1979 Sau1979]]


When 0 < a < 1, scaling is s = -1/2 - <span class="GramE">a,</span> and the following results are known:
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.


* LWP in <span class="SpellE">H^s</span> is known for s > 9/8 – 3a/8 [<span class="SpellE">KnKoe</span>-p]
** For s >= 3/4 (2-a) this is in [[references.html#KnPoVe1994b KnPoVe1994b]]
* GWP is known when s >= (a+1)/2 when a > 4/5, from the conservation of the Hamiltonian [[references.html#KnPoVe1994b KnPoVe1994b]]
* The LWP results are obtained by energy methods; it is known that pure iteration methods cannot work [[references.html#MlSauTz2001 MlSauTz2001]]
** 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]


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).The scaling exponent is 1/2 - 1<span class="GramE">/(</span>k-1).
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.


* 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 > ½ [<span class="SpellE">MlRi</span>-p]
==Estimates==
** For small data in <span class="SpellE">H^s</span>, s>1, LWP was obtained in [[references.html#KnPoVe1994b KnPoVe1994b]]
** 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.
** For s < ½, the solution map is not C^3 [<span class="SpellE">MlRi</span>-p]
* For k=4, LWP for small data in <span class="SpellE">H^s</span>, s > 5/6 was obtained in [[references.html#KnPoVe1994b KnPoVe1994b]].
* For k>4, LWP for small data in <span class="SpellE">H^s</span>, s >=3/4 was obtained in [[references.html#KnPoVe1994b KnPoVe1994b]].
* For any k >= 3 and s < 1/2 - 1/k the solution map is not uniformly continuous [<span class="SpellE">BiLi</span>-p]


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.It is globally well-posed in L^2 [[references.html#Li1999 Li1999]], 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).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].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">in</span> H^3 and above [[references.html#GuoTan1992 GuoTan1992]].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 [[references.html#OttSud1982 OttSud1982]].
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.


</div>
[[Category:Airy]]
[[Category:Equations]]

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.