# 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 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].
- 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
- 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

- 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 references:CoKeStTaTk-p3 CoKeStTkTa-p3
- Was shown for s >= 1 in St1997c
- Analytic well-posedness fails for s < 1/2 references:CoKeStTaTk-p3 CoKeStTkTa-p3, KnPoVe1996
- For arbitrary smooth non-linearities, weak H^1 solutions were constructed in references:Bo1993b Bo1993.

- On T with k >= 3, gKdV-k is GWP for s >= 1 except in the focussing case St1997c
- The estimates in references:CoKeStTaTk-p3 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 references:CoKeStTaTk-p3 CoKeStTkTa-p3. In the critical and super-critical cases there are some low-frequency issues which may require the techniques in [[references:KeTa-p KeTa-p]].