Generalized Korteweg-de Vries equation: Difference between revisions

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[Thanks to <span class="SpellE">Nikolaos</span> <span class="SpellE">Tzirakis</span> for some corrections - Ed.]
[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>[[Bibliography#KnPoVe1993|KnPoVe1993]]
** 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]]

Revision as of 19:48, 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 L^2 except in the critical case k=4 because one can scale solitons to be arbitrarily small in the non-critical cases.
    • Solitons are H^1-unstable BnSouSr1987
    • If one considers an arbitrary smooth non-linearity (not necessarily a power) then one has LWP for small data in H^s, s > 1/2 St1995
  • On R with any k, gKdV-k is GWP in H^s for s >= 1 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, gKdV-k has the H^s norm growing like t^{(s-1)+} 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 R^+ (or R^-) 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 O(t^{-1/3}) in L^\infty for small decaying data if k > (19 - sqrt(57))/4 ~ 2.8625... CtWs1991
    • A similar result for k > (5+sqrt(73))/4 ~ 3.39... 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