# Difference between revisions of "Generalized Korteweg-de Vries equation"

## Half-line theory

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

${\displaystyle \partial _{t}u+\partial _{x}^{3}u+\partial _{x}u+u^{k}\partial _{x}u=0;u(x,0)=u_{0}(x);u(0,t)=h(t)}$

The sign of ${\displaystyle \partial _{x}^{3}u}$ is important (it makes the influence of the boundary x=0 mostly negligible), the sign of ${\displaystyle u\partial _{x}u}$ is not. The drift term ${\displaystyle \partial _{x}u}$ is convenient for technical reasons; it is not known whether it is truly necessary.

• LWP is known for initial data in ${\displaystyle H^{s}}$ and boundary data in ${\displaystyle H^{(s+1)/3}}$ when ${\displaystyle s>3/4[CoKn-p]}$.
• The techniques are based on KnPoVe1993 and a replacement of the IVBP with a forced IVP.
• This has been improved to ${\displaystyle s>=\partial _{c}s=1/2-2/k}$when ${\displaystyle k>4[CoKe-p]}$.
• More specific results are known for KdV, mKdV, gKdV-3, and gKdV-4.

## Miscellaneous gKdV results

• On R with k > 4, ${\displaystyle gKdV-k}$is LWP down to scaling: ${\displaystyle s>=\partial _{c}s=1/2-2/k}$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 ${\displaystyle L^{2}}$except in the critical case k=4 because one can scale solitons to be arbitrarily small in the non-critical cases.
• Solitons are ${\displaystyle H^{1}}$-unstable BnSouSr1987
• If one considers an arbitrary smooth non-linearity (not necessarily a power) then one has LWP for small data in ${\displaystyle H^{s},s>1/2}$St1995
• On R with any k, gKdV-k is GWP in ${\displaystyle H^{s}}$ for s >= 1 KnPoVe1993, though for k >= 4 one needs the ${\displaystyle L^{2}}$norm to be small; global weak solutions were constructed much earlier, with the same smallness assumption when k >= 4. This should be improvable below ${\displaystyle H^{1}}$for all k.
• On R with any k, gKdV-k has the ${\displaystyle H^{s}}$ norm growing like ${\displaystyle t^{(s-1)}}$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 ${\displaystyle R^{+}}$(or ${\displaystyle R^{-}}$) for two different times KnPoVe-p3, [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 ${\displaystyle O(t^{-1/3})inL^{\infty }}$ for small decaying data if ${\displaystyle k>{(19-{\sqrt {(}}57)) \over 4}\sim 2.8625...}$ CtWs1991
• A similar result for ${\displaystyle k>(5+{\sqrt {(}}73))/4\sim 3.39...}$was obtained in PoVe1990.
• When k=2 solutions decay like ${\displaystyle O(t^{-1/3})}$, and when k=1 solutions decay generically like ${\displaystyle O(t^{-2/3})}$ but like ${\displaystyle O((t/logt)^{-2/3})}$ for exceptional data AbSe1977
• In the ${\displaystyle L^{2}}$ subcritical case 0 < k < 4, multisoliton solutions are asymptotically ${\displaystyle H^{1}}$-stable [MtMeTsa-p]
• A dissipative version of gKdV-k was analyzed in MlRi2001
• On T with any k, gKdV-k has the ${\displaystyle H^{s}}$ norm growing like ${\displaystyle t^{2(s-1)+}}$ 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 ${\displaystyle H^{1}}$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.