Generalized Korteweg-de Vries equation

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Half-line theory

The gKdV Cauchy-boundary problem on the half-line is

u_t + u_{xxx} + u_x + u^k u_x = 0; u(x,0) = u_0(x); u(0,t) = h(t)

The sign of u_{xxx} is important (it makes the influence of the boundary x=0 mostly negligible), the sign of u u_x is not. The drift term u_x is convenient for technical reasons; it is not known whether it is truly necessary.

  • LWP is known for initial data in H^s and boundary data in H^{(s+1)/3} when s > 3/4 [CoKn-p].
    • The techniques are based on KnPoVe1993 and a replacement of the IVBP with a forced IVP.
    • This has been improved to s >= s_c = 1/2 - 2/k when k > 4 [CoKe-p].
    • 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, gKdV-k is LWP down to scaling: s >= s_c = 1/2 - 2/k 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