Cubic NLS on T
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Description | |
---|---|
Equation | |
Fields | |
Data class | |
Basic characteristics | |
Structure | completely integrable |
Nonlinearity | semilinear |
Linear component | Schrodinger |
Critical regularity | |
Criticality | mass-subcritical; energy-subcritical |
Covariance | Galilean |
Theoretical results | |
LWP | for |
GWP | for |
Related equations | |
Parent class | cubic NLS |
Special cases | - |
Other related | KdV, mKdV |
The theory of the cubic NLS on the circle is as follows.
- LWP for Bo1993.
- GWP for thanks to conservation Bo1993.
- One also has GWP for random data whose Fourier coefficients decay like (times a Gaussian random variable) Bo1995c. Indeed one has an invariant measure.
- If the cubic non-linearity is of type (instead of ) then one can obtain LWP for Gr-p2
- Remark: This equation is completely integrable AbMa1981; all higher order integer Sobolev norms stay bounded. Growth of fractional norms might be interesting, though.