View source for Generalized Korteweg-de Vries equation

== Half-line theory ==

The <span class="SpellE">gKdV</span> Cauchy-boundary problem on the half-line is

<center><math>\partial_t u + \partial_x^3 u + \partial_x u + u^k \partial_x u = 0; u(x,0) = u_0(x); u(0,t) = h(t)</math></center>

The sign of <math>\partial_x^3 u </math> is important (it makes the influence of the boundary x=0 mostly negligible), the sign of <math>u \partial_x u</math> is not. The drift term <math>\partial_x u</math> is convenient for technical reasons; it is not known whether it is truly necessary.

* LWP is known for initial data in <math>H^s</math> and boundary data in <math>H^{(s+1)/3}</math> when <math>s > 3/4 [CoKn-p]</math>.
** The techniques are based on [[Bibliography#KnPoVe1993|KnPoVe1993]] and a replacement of the IVBP with a forced IVP.
** This has been improved to <math>s >= \partial_c s = 1/2 - 2/k </math>when <math>k > 4 [CoKe-p]</math>.
** More specific results are known for [[Korteweg-de Vries on the half-line|KdV]], [[modified Korteweg-de Vries on the half-line|mKdV]], [[gKdV-3 on the half-line|gKdV-3]], and [[gKdV-4 on the half-line|gKdV-4]].

== Miscellaneous gKdV results ==

[Thanks to <span class="SpellE">Nikolaos</span> <span class="SpellE">Tzirakis</span> for some corrections - Ed.]

* On R with k > 4, <math>gKdV-k </math>is LWP down to scaling: <math>s >= \partial_c s = 1/2 - 2/k </math>[[Bibliography#KnPoVe1993|KnPoVe1993]]
** Was shown for s>3/2 in [[Bibliography#GiTs1989|GiTs1989]]
** One has ill-<span class="SpellE">posedness</span> in the supercritical regime [[Bibliography#BirKnPoSvVe1996|BirKnPoSvVe1996]]
** For small data one has scattering [[Bibliography#KnPoVe1993c|KnPoVe1993c]].Note that one cannot have scattering in <math>L^2 </math>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 <math>H^1</math>-unstable [[Bibliography#BnSouSr1987|BnSouSr1987]]
** If one considers an arbitrary smooth non-linearity (not necessarily a power) then one has LWP for small data in <math>H^s, s > 1/2 </math>[[Bibliography#St1995|St1995]]
* On R with any k, <span class="SpellE">gKdV</span>-k is GWP in <math>H^s</math> for s >= 1 [[Bibliography#KnPoVe1993|KnPoVe1993]], though for k >= 4 one needs the <math>L^2 </math>norm to be small; global weak solutions were constructed much earlier, with the same smallness assumption when k >= 4. This should be improvable below <math>H^1 </math>for all k.
* On R with any k, <span class="SpellE">gKdV</span>-k has the <math>H^s</math> norm growing like <math>t^{(s-1)} </math>in time for any integer s >= 1 [[Bibliography#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 <math>R^+ </math>(or <math>R^-</math>) for two different times [[references:KnPoVe-p3 KnPoVe-p3]], [KnPoVe-p4].
** In the completely <span class="SpellE">integrable</span> cases k=1,2 this is in [[Bibliography#Zg1992|Zg1992]]
** Also, a non-zero solution to <span class="SpellE">gKdV</span> cannot vanish on a rectangle in <span class="SpellE">spacetime</span> [[Bibliography#SauSc1987|SauSc1987]]; see also [[Bibliography#Bo1997b|Bo1997b]].
** Extensions to higher order <span class="SpellE">gKdV</span> type equations are in [[Bibliography#Bo1997b|Bo1997b]], [KnPoVe-p5].
* On R with non-integer k, one has decay of <math>O(t^{-1/3}) in L^\infty</math> for small decaying data if <math>k > (19 - \sqrt(57))/4 \sim 2.8625...</math> [[Bibliography#CtWs1991|CtWs1991]]
** A similar result for <math> k > (5+\sqrt(73))/4 \sim 3.39... </math><span class="GramE">was</span> obtained in [[Bibliography#PoVe1990|PoVe1990]].
** When k=2 solutions decay like <math>O(t^{-1/3})</math>, and when k=1 solutions decay generically like <math>O(t^{-2/3})</math> but like <math>O( (t/log t)^{-2/3})</math> for exceptional data [[Bibliography#AbSe1977|AbSe1977]]
* In the <math>L^2</math> <span class="SpellE">subcritical</span> case 0 < k < 4, <span class="SpellE">multisoliton</span> solutions are asymptotically <math>H^1</math>-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], [[Bibliography#Miz2001|Miz2001]]; earlier work is in [[Bibliography#Bj1972|Bj1972]], [[Bibliography#Bn1975|Bn1975]], [[Bibliography#Ws1986|Ws1986]], [[Bibliography#PgWs1994|PgWs1994]]
* A dissipative version of <span class="SpellE">gKdV</span>-k was analyzed in [[Bibliography#MlRi2001|MlRi2001]]

* On T with any k, <span class="SpellE">gKdV</span>-k has the <math>H^s</math> norm growing like <math>t^{2(s-1)+}</math> in time for any integer s >= 1 [[Bibliography#St1997b|St1997b]]
* On T with k >= 3, <span class="SpellE">gKdV</span>-k is LWP for s >= 1/2 [[references:CoKeStTaTk-p3 CoKeStTkTa-p3]]
** Was shown for s >= 1 in [[Bibliography#St1997c|St1997c]]
** Analytic well-<span class="SpellE">posedness</span> fails for s < 1/2 [[references:CoKeStTaTk-p3 CoKeStTkTa-p3]], [[Bibliography#KnPoVe1996|KnPoVe1996]]
** For arbitrary smooth non-<span class="SpellE">linearities</span>, weak <math>H^1 </math>solutions were constructed in [[references: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 [[Bibliography#St1997c|St1997c]]
** The estimates in [[references: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:CoKeStTaTk-p3 CoKeStTkTa-p3]]. In the critical and super-critical cases there are some low-frequency issues which may require the techniques in [[references:KeTa-p <span class="SpellE">KeTa</span>-p]].

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