GKdV-3 equation
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Description | |
---|---|
Equation | |
Fields | |
Data class | |
Basic characteristics | |
Structure | Hamiltonian |
Nonlinearity | semilinear with derivatives |
Linear component | Airy |
Critical regularity | |
Criticality | mass-subcritical, energy-subcritical |
Covariance | - |
Theoretical results | |
LWP | for |
GWP | for , small norm |
Related equations | |
Parent class | gKdV |
Special cases | - |
Other related | - |
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 for s >= -1/6 Ta2007
- For s > -1/6 this is in 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 CoKn-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 CoKn-p
- Solitons are H^1-stable CaLo1982, Ws1986, BnSouSr1987 and asymptotically H^1 stable MtMe-p3, MtMe-p
- If one also assumes the error is small in the critical space then one has asymptotic stability Ta2007
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 correction term method.
- Remark: For this equation it is convenient to make a gauge transformation to subtract off the mean of P(u).