Generalized Korteweg-de Vries equation: Difference between revisions

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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.
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 <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].
* 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</math> [[CoKn-p]].
** The techniques are based on [[Bibliography#KnPoVe1993|KnPoVe1993]] and a replacement of the IVBP with a forced IVP.
** The techniques are based on [[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].
** This has been improved to <math>s >= \partial_c s = 1/2 - 2/k </math>when <math>k > 4</math> [[CoKn-p]].
** 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]].
** 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 ==
== Miscellaneous gKdV results ==


[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 [[Bibliography#KnPoVe1993|KnPoVe1993]]
* On R with k > 4, <math>gKdV-k </math>is LWP down to scaling: <math>s >= \partial_c s = 1/2 - 2/k </math>[[KnPoVe1993]]
** Was shown for s>3/2 in [[Bibliography#GiTs1989|GiTs1989]]
** Was shown for s>3/2 in [[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 [[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 [[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 [[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>[[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 [[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 [[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 [[KnPoVe2003]], [[KnPoVe-p4]].
** In the completely <span class="SpellE">integrable</span> cases k=1,2 this is in [[Bibliography#Zg1992|Zg1992]]
** In the [[completely integrable]] cases k=1,2 this is in [[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> [[SauSc1987]]; see also [[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 [[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(t^{-1/3}) in L^\infty</math> for small decaying data if <math>k > {(19 - \sqrt(57)) \over 4} \sim 2.8625...</math> [[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 \sim 3.39... </math><span class="GramE">was</span> 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 [[Bibliography#AbSe1977|AbSe1977]]
** 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 [[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 <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 [[MtMeTsa-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]]
** For a single <span class="SpellE">soliton</span> this is in [[MtMe-p3]], [[MtMe-p]], [[Miz2001]]; earlier work is in [[Bj1972]], [[Bn1975]], [[Ws1986]], [[PgWs1994]]
* A dissipative version of <span class="SpellE">gKdV</span>-k was analyzed in [[Bibliography#MlRi2001|MlRi2001]]
* A dissipative version of <span class="SpellE">gKdV</span>-k was analyzed in [[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 [[Bibliography#St1997b|St1997b]]
* 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 [[St1997b]]
* On T with k >= 3, <span class="SpellE">gKdV</span>-k is LWP for s >= 1/2 [[references:CoKeStTaTk-p3 CoKeStTkTa-p3]]
* On T with k >= 3, <span class="SpellE">gKdV</span>-k is LWP for s >= 1/2 [[CoKeStTkTa-p3]]
** Was shown for s >= 1 in [[Bibliography#St1997c|St1997c]]
** Was shown for s >= 1 in [[St1997c]]
** Analytic well-<span class="SpellE">posedness</span> fails for s < 1/2 [[references:CoKeStTaTk-p3 CoKeStTkTa-p3]], [[Bibliography#KnPoVe1996|KnPoVe1996]]
** Analytic well-<span class="SpellE">posedness</span> fails for s < 1/2 [[CoKeStTkTa-p3]], [[KnPoVe1996]]
** For arbitrary smooth non-<span class="SpellE">linearities</span>, weak H^1 solutions were constructed in [[references:Bo1993b Bo1993]].
** For arbitrary smooth non-<span class="SpellE">linearities</span>, weak <math>H^1 </math>solutions were constructed in [[Bo1993b]].
* 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]]
* 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 [[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]].
** The estimates in [[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 [[CoKeStTkTa-p3]]. In the critical and super-critical cases there are some low-frequency issues which may require the techniques in [[KeTa-p]].


[[Category:Equations]]
[[Category:Equations]]
[[Category:Airy]]

Latest revision as of 20:49, 10 June 2007

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

Miscellaneous gKdV results

  • 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 KnPoVe2003, KnPoVe-p4.
  • 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 , and when k=1 solutions decay generically like but like for exceptional data AbSe1977
  • In the subcritical case 0 < k < 4, multisoliton solutions are asymptotically -stable MtMeTsa-p
  • A dissipative version of gKdV-k was analyzed in MlRi2001
  • On T with any k, gKdV-k has the norm growing like in time for any integer s >= 1 St1997b
  • On T with k >= 3, gKdV-k is LWP for s >= 1/2 CoKeStTkTa-p3
    • Was shown for s >= 1 in St1997c
    • Analytic well-posedness fails for s < 1/2 CoKeStTkTa-p3, KnPoVe1996
    • For arbitrary smooth non-linearities, weak solutions were constructed in Bo1993b.
  • On T with k >= 3, gKdV-k is GWP for s >= 1 except in the focussing case St1997c
    • The estimates in 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 CoKeStTkTa-p3. In the critical and super-critical cases there are some low-frequency issues which may require the techniques in KeTa-p.