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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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 $k>4[CoKe-p]$.
  • 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: $s>=\partial _{c}s=1/2-2/k$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 $L^{2}$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].
  • 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 $O(t^{-1/3})inL^{\infty }$ for small decaying data if $k>{(19-{\sqrt {(}}57)) \over 4}\sim 2.8625...$ CtWs1991
  • A similar result for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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], 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\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 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]].