Difference between revisions of "Generalized Korteweg-de Vries equation"

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* 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 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 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].
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* 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 [[Bibliography#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]].
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* 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 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]]
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* On T with k >= 3, <span class="SpellE">gKdV</span>-k is LWP for s >= 1/2 [[Bibliography#CoKeStTaTk-p3 |CoKeStTkTa-p3]]
 
** Was shown for s >= 1 in [[Bibliography#St1997c|St1997c]]
 
** 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]]
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** Analytic well-<span class="SpellE">posedness</span> fails for s < 1/2 [[Bibliography#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]].
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** For arbitrary smooth non-<span class="SpellE">linearities</span>, weak <math>H^1 </math>solutions were constructed in [[Bibliography#Bo1993b |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 [[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]].
+
** The estimates in [[Bibliography#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 [[Bibliography#CoKeStTaTk-p3 |CoKeStTkTa-p3]]. In the critical and super-critical cases there are some low-frequency issues which may require the techniques in [[Bibliography#KeTa-p |KeTa-p]].
  
 
[[Category:Equations]]
 
[[Category:Equations]]
 
[[Category:Airy]]
 
[[Category:Airy]]

Revision as of 16:45, 31 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

  • 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 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 , 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.