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 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 when k > 4 CoKn-p.
  • More specific results are known for KdV, mKdV, gKdV-3, and gKdV-4.

Miscellaneous gKdV results

• On R with k > 4, gKdV − kis 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 L²except in the critical case k=4 because one can scale solitons to be arbitrarily small in the non-critical cases.
  • Solitons are H¹-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 / 2St1995
• On R with any k, gKdV-k is GWP in H^s for s >= 1 KnPoVe1993, though for k >= 4 one needs the L²norm to be small; global weak solutions were constructed much earlier, with the same smallness assumption when k >= 4. This should be improvable below H¹for 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 KnPoVe2003, 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 O(t ^{− 1 / 3}), and when k=1 solutions decay generically like O(t ^{− 2 / 3}) but like O((t / logt) ^{− 2 / 3}) for exceptional data AbSe1977
• In the L² subcritical case 0 < k < 4, multisoliton solutions are asymptotically H¹-stable MtMeTsa-p
  • For a single soliton this is in MtMe-p3, MtMe-p, Miz2001; earlier work is in Bj1972, Bn1975, Ws1986, PgWs1994
• 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 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 H¹solutions 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.