Generalized Korteweg-de Vries equation: Difference between revisions

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** Was shown for s>3/2 in [[Bibliography#GiTs1989|GiTs1989]]
** 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]]
** 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 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.
** 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 H^1-unstable [[Bibliography#BnSouSr1987|BnSouSr1987]]
** <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 <span class="SpellE">H^s</span>, s > 1/2 [[Bibliography#St1995|St1995]]
** 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 <span class="SpellE">H^s</span> for s >= 1 [[Bibliography#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 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 <span class="SpellE">H^s</span> norm growing like t^{(s-1)+} in time for any integer s >= 1 [[Bibliography#St1997b|St1997b]]
* 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 R^+ (or R^-) for two different times [[references:KnPoVe-p3 KnPoVe-p3]], [KnPoVe-p4].
* 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]]
** 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]].
** 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].
** 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 <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... [[Bibliography#CtWs1991|CtWs1991]]
* On R with non-integer k, one has decay of <math>O(</span>t^{-1/3}) in L^\infty</math> for small decaying data if <math>k > (19 - sqrt(57))/4 ~ 2.8625...</math> [[Bibliography#CtWs1991|CtWs1991]]
** A similar result for k > (5+<span class="GramE">sqrt(</span>73))/4 ~ 3.39... <span class="GramE">was</span> obtained in [[Bibliography#PoVe1990|PoVe1990]].
** A similar result for <math> k > (5+sqrt(73))/4 ~ 3.39... </math><span class="GramE">was</span> obtained in [[Bibliography#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 [[Bibliography#AbSe1977|AbSe1977]]
** 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 [[Bibliography#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]
* 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]

Revision as of 19:52, 28 July 2006

Half-line theory

The gKdV Cauchy-boundary problem on the half-line is

The sign of is important (it makes the influence of the boundary x=0 mostly negligible), the sign of is not. The drift term is convenient for technical reasons; it is not known whether it is truly necessary.

  • LWP is known for initial data in and boundary data in when .
    • The techniques are based on KnPoVe1993 and a replacement of the IVBP with a forced IVP.
    • This has been improved to when .
    • More specific results are known for KdV, mKdV, gKdV-3, and gKdV-4.

Miscellaneous gKdV results

[Thanks to Nikolaos Tzirakis for some corrections - Ed.]

  • On R with k > 4, is LWP down to scaling: KnPoVe1993
    • Was shown for s>3/2 in GiTs1989
    • One has ill-posedness in the supercritical regime BirKnPoSvVe1996
    • For small data one has scattering KnPoVe1993c.Note that one cannot have scattering in except in the critical case k=4 because one can scale solitons to be arbitrarily small in the non-critical cases.
    • Solitons are -unstable BnSouSr1987
    • If one considers an arbitrary smooth non-linearity (not necessarily a power) then one has LWP for small data in St1995
  • On R with any k, gKdV-k is GWP in for s >= 1 KnPoVe1993, though for k >= 4 one needs the norm to be small; global weak solutions were constructed much earlier, with the same smallness assumption when k >= 4. This should be improvable below for all k.
  • On R with any k, gKdV-k has the norm growing like in time for any integer s >= 1 St1997b
  • On R with any non-linearity, a non-zero solution to gKdV cannot be supported on the half-line (or ) for two different times references:KnPoVe-p3 KnPoVe-p3, [KnPoVe-p4].
    • In the completely integrable cases k=1,2 this is in Zg1992
    • Also, a non-zero solution to gKdV cannot vanish on a rectangle in spacetime SauSc1987; see also Bo1997b.
    • Extensions to higher order gKdV type equations are in Bo1997b, [KnPoVe-p5].
  • On R with non-integer k, one has decay of for small decaying data if CtWs1991
    • A similar result for was obtained in 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 AbSe1977
  • In the L^2 subcritical case 0 < k < 4, multisoliton solutions are asymptotically H^1-stable [MtMeTsa-p]
  • A dissipative version of gKdV-k was analyzed in MlRi2001