Cubic NLS on R

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Cubic NLS on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \R}
Description
Equation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle iu_t + u_{xx} = \pm |u|^2 u}
Fields Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u: \R \times \R \to \mathbb{C}}
Data class Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u(0) \in H^s(\R)}
Basic characteristics
Structure completely integrable
Nonlinearity semilinear
Linear component Schrodinger
Critical regularity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \dot H^{-1/2}(\R)}
Criticality mass-subcritical;
energy-subcritical;
scattering-critical
Covariance Galilean
Theoretical results
LWP Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^s(\R)} for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s \geq 0}
GWP Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^s(\R)} for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s \geq 0}
Related equations
Parent class cubic NLS
Special cases -
Other related KdV, mKdV


The theory of the cubic NLS on one dimension is as follows.

  • LWP for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s \ge 0\,} Ts1987, CaWe1990 (see also GiVl1985).
    • This is sharp for reasons of Gallilean invariance and for soliton solutions in the focussing case KnPoVe2001
      • The result is also sharp in the defocussing case CtCoTa-p, due to Gallilean invariance and the asymptotic solutions in Oz1991.
      • Below Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s \ge 0\,} the solution map was known to be not Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle C^2\,} in Bo1993
    • For initial data equal to a delta function there are serious problems with existence and uniqueness KnPoVe2001.
    • However, there exist Gallilean invariant spaces which scale below Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L^2\,} for which one has LWP. They are defined in terms of the Fourier transform VaVe2001. For instance one has LWP for data whose Fourier transform decays like Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle |x|^{-1/6-}\,} . Ideally one would like to replace this with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle |x|^{0-}\,.}
  • GWP for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s \ge 0\,} thanks to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L^2\,} conservation.
    • GWP can be pushed below to certain of the Gallilean spaces in VaVe2001. For instance one has GWP when the Fourier transform of the data decays like Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle |x|^{-5/12-}\,} . Ideally one would like to replace this with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 0-} .
  • If the cubic non-linearity is of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \underline{uuu}\,} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u u u\,} type (as opposed to the usual Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle |u|^2 u\,} type) then one can obtain LWP for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s > -5/12\,} Gr-p2. If the nonlinearity is of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \underline{uu} u\,} type then one has LWP for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s > -2/5\,} Gr-p2.
  • Remark: This equation is sometimes known as the Zakharov-Shabat equation and is completely integrable (see e.g. AbKauNeSe1974; all higher order integer Sobolev norms stay bounded. Growth of fractional norms might be interesting, though.
  • In the focusing case there are soliton and multisoliton solutions, however the defocusing case does not admit such solutions.
  • In the focussing case there is a unique positive radial ground state for each energy Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle E\,} . By translation and phase shift one thus obtains a four-dimensional manifold of ground states (aka solitons) for each energy. This manifold is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^1\,} -stable Ws1985, Ws1986. Below the energy norm orbital stability is not known, however there are polynomial bounds on the instability CoKeStTkTa2003b.
  • This equation is related to the evolution of vortex filaments under the localized induction approximation, via the Hasimoto transformation, see e.g. Hm1972
  • Solutions do not scatter to free Schrodinger solutions. In the focussing case this can be easily seen from the existence of solitons. But even in the defocussing case wave operators do not exist, and must be replaced by modified wave operators Oz1991, see also CtCoTa-p. For small, decaying data one also has asymptotic completeness HaNm1998.
    • For large Schwartz data, these asymptotics can be obtained by inverse scattering methods ZkMan1976, SeAb1976, No1980, DfZx1994
    • For large real analytic data, these asymptotics were obtained in GiVl2001
    • Refinements to the convergence and regularity of the modified wave operators was obtained in Car2001.

Cubic NLS on the half-line and interval

  • On the half line Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle R^+\,} , global well-posedness in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^2\,} was established in CrrBu1991, Bu.1992
  • On the interval, the inverse scattering method was applied to generate solutions in GriSan-p.
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