Cubic NLS on
Description
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Equation
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Fields
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Data class
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Basic characteristics
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Structure
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completely integrable
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Nonlinearity
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semilinear
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Linear component
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Schrodinger
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Critical regularity
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Criticality
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mass-subcritical; energy-subcritical; scattering-critical
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Covariance
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Galilean
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Theoretical results
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LWP
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for
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GWP
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for
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Related equations
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Parent class
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cubic NLS
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Special cases
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-
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Other related
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KdV, mKdV
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The theory of the cubic NLS on the circle is as follows.
- LWP for
Bo1993.
- For
one has failure of uniform local well-posedness CtCoTa-p, BuGdTz-p. In fact, the solution map is not even continuous from
to
for any
, even for small times and small data CtCoTa-p3.
- 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.