# Generalized Korteweg-de Vries equation

## Half-line theory

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

$\partial_t u + \partial_x^3 u + \partial_x u + u^k \partial_x u = 0; u(x,0) = u_0(x); u(0,t) = h(t)$

The sign of $\partial_x^3 u$ is important (it makes the influence of the boundary x=0 mostly negligible), the sign of $u \partial_x u$ is not. The drift term $\partial_x u$ 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.
• The techniques are based on KnPoVe1993 and a replacement of the IVBP with a forced IVP.
• This has been improved to $s >= \partial_c s = 1/2 - 2/k$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, gKdVkis LWP down to scaling: $s >= \partial_c s = 1/2 - 2/k$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 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 $O(t^{-1/3}) in L^\infty$ for small decaying data if $k > {(19 - \sqrt(57)) \over 4} \sim 2.8625...$ CtWs1991
• A similar result for $k > (5+\sqrt(73))/4 \sim 3.39...$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 / logt) − 2 / 3) for exceptional data AbSe1977
• 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
• 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.