Generalized Korteweg-de Vries equation

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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.