Generalized Korteweg-de Vries equation
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 Hs and boundary data in H(s + 1) / 3 when s > 3 / 4 CoKn-p.
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 Hs,s > 1 / 2St1995
- On R with any k, gKdV-k is GWP in Hs 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 Hs 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.
- On R with non-integer k, one has decay of for small decaying data if CtWs1991
- In the L2 subcritical case 0 < k < 4, multisoliton solutions are asymptotically H1-stable MtMeTsa-p
- A dissipative version of gKdV-k was analyzed in MlRi2001
- On T with any k, gKdV-k has the Hs norm growing like t2(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
- On T with k >= 3, gKdV-k is GWP for s >= 1 except in the focussing case St1997c