Modified Korteweg-de Vries on R

The local and global well-posedness theory for the modified Korteweg-de Vries equation on the line and half-line is as follows.

• Scaling is s_c = -1/2.
• LWP in H^s for s >= 1/4 references.html#KnPoVe1993 KnPoVe1993
  • Was shown for s>3/2 in references.html#GiTs1989 GiTs1989
  • This is sharp in the focussing case [[references.html#KnPoVe-p KnPoVe-p]], in the sense that the solution map is no longer uniformly continuous for s < 1/4.
    • This has been extended to the defocussing case in [CtCoTa-p], by a high-frequency approximation of mKdV by [schrodinger.html#Cubic NLS on R NLS]. (This high frequency approximation has also been utilized in references.html#Sch1998 Sch1998).
    • Below 1/4 the solution map was known to not be C^3 in references.html#Bo1993b Bo1993b, references.html#Bo1997 Bo1997.
  • The same result has also been established for the half-line [CoKe-p], assuming boundary data is in H^{(s+1)/3} of course.
  • Global weak solutions in L^2 were constructed in references.html#Ka1983 Ka1983. Thus in L^2 one has global existence but no uniform continuity.
  • Uniqueness is also known when the initial data lies in the weighted space <x>^{3/8} u_0 in L^2 references.html#GiTs1989 GiTs1989
  • LWP has also been demonstrated when <xi>^s hat(u_0) lies in L^{r’} for 4/3 < r <= 2 and s >= ½ - 1/2r [Gr-p4]
• GWP in H^s for s > 1/4 references.html#CoKeStTaTk-p2 CoKeStTkTa-p2, via the KdV theory and the Miura transform, for both the focussing and defocussing cases.
  • Was proven for s>3/5 in references.html#FoLiPo1999 FoLiPo1999
  • Is implicit for s >= 1 from references.html#KnPoVe1993 KnPoVe1993
  • On the half-line GWP is known when s >= 1 and the boundary data is in H^{11/12}, assuming compatibility and small L^2 norm [CoKe-p]
  • GWP for smooth data can also be achieved from inverse scattering methods [BdmFsShp-p]; the same approach also works on an interval [BdmShp-p].
  • Solitons are asymptotically H^1 stable [MtMe-p3], [MtMe-p]