Difference between revisions of "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 KnPoVe1993
• Was shown for s>3/2 in GiTs1989
• This is sharp in the focusing case KnPoVe2001, in the sense that the solution map is no longer uniformly continuous for s < 1/4.
• This has been extended to the defocusing case in CtCoTa-p, by a high-frequency approximation of mKdV by cubic NLS. (This high frequency approximation has also been utilized in Sch1998).
• Below 1/4 the solution map was known to not be C^3 in Bo1993b, Bo1997.
• The same result has also been established for the half-line CoKn-p, assuming boundary data is in H^{(s+1)/3} of course.
• Global weak solutions in L^2 were constructed in 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 GiTs1989
• LWP has also been demonstrated when ${\displaystyle <\xi >^{s}{\hat {u_{0}}}}$ lies in ${\displaystyle L^{r/(r-1)}}$ for 4/3 < r <= 2 and s >= ½ - 1/2r Gr-p4
• GWP in H^s for s > 1/4 CoKeStTkTa2003, via the KdV theory and the Miura transform, for both the focussing and defocussing cases.
• Was proven for s>3/5 in FoLiPo1999
• Is implicit for s >= 1 from 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 CoKn-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