Generalized Korteweg-de Vries equation

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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 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 $s >= \partial_c s = 1/2 - 2/k$ when $k > 4$ CoKn-p.
- More specific results are known for KdV, mKdV, gKdV-3, and gKdV-4.

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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 $L 2$ 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 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 $k > (5+\sqrt(73))/4 \sim 3.39...$ 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 $O ((t / l o g t) - 2 / 3)$ 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 $H 1$ solutions 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.