GKdV-3 equation
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Non-periodic theory
The local and global well-posedness theory for the quartic generalized Korteweg-de Vries equation on the line and half-line is as follows.
- Scaling is s_c = -1/6.
- LWP in H^s for s > -1/6 [Gr-p3]
- Was shown for s>=1/12 KnPoVe1993
- Was shown for s>3/2 in GiTs1989
- The result s >= 1/12 has also been established for the half-line [CoKe-p], assuming boundary data is in H^{(s+1)/3} of course..
- GWP in H^s for s >= 0 [Gr-p3]
- For s>=1 this is in KnPoVe1993
- Presumably one can use either the Fourier truncation method or the "I-method" to go below L^2. Even though the equation is not completely integrable, the one-dimensional nature of the equation suggests that "correction term" techniques will also be quite effective.
- On the half-line GWP is known when s >= 1 and the boundary data is in H^{5/4}, assuming compatibility and small L^2 norm [CoKe-p]
- Solitons are H^1-stable CaLo1982, Ws1986, BnSouSr1987 and asymptotically H^1 stable [MtMe-p3], [MtMe-p]
Periodic theory
The local and global well-posedness theory for the quartic generalized Korteweg-de Vries equation on the line and half-line is as follows.
- Scaling is s_c = -1/6.
- LWP in H^s for s>=1/2 CoKeStTkTa-p3
- Was shown for s >= 1 in St1997c
- One has analytic ill-posedness for s<1/2 CoKeStTkTa-p3 by a modification of the example in KnPoVe1996.
- GWP in H^s for s>5/6 CoKeStTkTa-p3
- Was shown for s >= 1 in St1997c
- This result may well be improvable by the "damping correction term" method in [[Bibliography#CoKeStTaTk-p2 |CoKeStTkTa-p2]].
- Remark: For this equation it is convenient to make a "gauge transformation'' to subtract off the mean of P(u).
