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 < | ** 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 < | * 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 < | * 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 < | * 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+ | ** 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].
- On R with non-integer k, one has decay of for small decaying data if CtWs1991
- 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
- On T with any k, gKdV-k has the H^s norm growing like t^{2(s-1)+} in time for any integer s >= 1 St1997b
- On T with k >= 3, gKdV-k is LWP for s >= 1/2 references:CoKeStTaTk-p3 CoKeStTkTa-p3
- Was shown for s >= 1 in St1997c
- Analytic well-posedness fails for s < 1/2 references:CoKeStTaTk-p3 CoKeStTkTa-p3, KnPoVe1996
- For arbitrary smooth non-linearities, weak H^1 solutions were constructed in references:Bo1993b Bo1993.
- On T with k >= 3, gKdV-k is GWP for s >= 1 except in the focussing 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 KeTa-p]].