# Modified Korteweg-de Vries on R

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Jump to navigationJump to searchThe 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 focussing case KnPoVe-p, in the sense that the solution map is no longer uniformly continuous for s < 1/4.
- 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 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 <xi>^s hat(u_0) lies in L^{r\u2019} for 4/3 < r <= 2 and s >= ½ - 1/2r [Gr-p4]

- GWP in H^s for s > 1/4 CoKeStTkTa-p2, 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 [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]