# Generalized Korteweg-de Vries equation

From DispersiveWiki

Jump to navigationJump to search## 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 references.html#KnPoVe1993 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 references.html#KnPoVe1993 KnPoVe1993
- Was shown for s>3/2 in references.html#GiTs1989 GiTs1989
- One has ill-posedness in the supercritical regime references.html#BirKnPoSvVe1996 BirKnPoSvVe1996
- For small data one has scattering references.html#KnPoVe1993c 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 references.html#BnSouSr1987 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 references.html#St1995 St1995

- On R with any k, gKdV-k is GWP in H^s for s >= 1 references.html#KnPoVe1993 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 references.html#St1997b 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.html#KnPoVe-p3 KnPoVe-p3, [KnPoVe-p4].
- In the completely integrable cases k=1,2 this is in references.html#Zg1992 Zg1992
- Also, a non-zero solution to gKdV cannot vanish on a rectangle in spacetime references.html#SauSc1987 SauSc1987; see also references.html#Bo1997b Bo1997b.
- Extensions to higher order gKdV type equations are in references.html#Bo1997b 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... references.html#CtWs1991 CtWs1991
- A similar result for k > (5+sqrt(73))/4 ~ 3.39... was obtained in references.html#PoVe1990 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 references.html#AbSe1977 AbSe1977

- In the L^2 subcritical case 0 < k < 4, multisoliton solutions are asymptotically H^1-stable [MtMeTsa-p]
- For a single soliton this is in [MtMe-p3], [MtMe-p], references.html#Miz2001 Miz2001; earlier work is in references.html#Bj1972 Bj1972, references.html#Bn1975 Bn1975, references.html#Ws1986 Ws1986, references.html#PgWs1994 PgWs1994

- A dissipative version of gKdV-k was analyzed in references.html#MlRi2001 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 references.html#St1997b St1997b
- On T with k >= 3, gKdV-k is LWP for s >= 1/2 references.html#CoKeStTaTk-p3 CoKeStTkTa-p3
- Was shown for s >= 1 in references.html#St1997c St1997c
- Analytic well-posedness fails for s < 1/2 references.html#CoKeStTaTk-p3 CoKeStTkTa-p3, references.html#KnPoVe1996 KnPoVe1996
- For arbitrary smooth non-linearities, weak H^1 solutions were constructed in references.html#Bo1993b Bo1993.

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