On the one-dimensional cubic nonlinear Schrodinger equation below L^2

Abstract

In this paper, we review several recent results concerning well-posedness of the one-dimensional, cubic Nonlinear Schrodinger equation (NLS) on the real line R and on the circle T for solutions below the L^2-threshold. We point out common results for NLS on R and the so-called "Wick ordered NLS" (WNLS) on T, suggesting that WNLS may be an appropriate model for the study of solutions below L^2(T). In particular, in contrast with a recent result of Molinet who proved that the solution map for the periodic cubic NLS equation is not weakly continuous from L^2(T) to the space of distributions, we show that this is not the case for WNLS.

14 pages, additional references

References in corpus (7)

• Almost sure well-posedness of the cubic nonlinear Schrödinger equation below L^2(T)
• Low regularity local well-posedness of the Derivative Nonlinear Schrödinger Equation with periodic initial data
• Instability of the periodic nonlinear Schrodinger equation
• Nonuniqueness of weak solutions of the nonlinear Schroedinger equation
• Normal form theory for the NLS equation
• Interpolation of Gibbs measures with White Noise for Hamiltonian PDE
• Weak Continuity of Dynamical Systems for the KdV and mKdV Equations

Cited by in corpus (16)

• Almost sure well-posedness of the cubic nonlinear Schrödinger equation below L^2(T)
• Randomization and the Gross-Pitaevskii hierarchy
• Solving the 4NLS with white noise initial data
• Unconditional Uniqueness Results for the Nonlinear Schrödinger Equation
• Global well-posedness of the one-dimensional cubic nonlinear Schrödinger equation in almost critical spaces
• Local existence of solutions to Randomized Gross-Pitaevskii hierarchies
• On the global well-posedness of the Calogero-Sutherland derivative nonlinear Schrödinger equation
• Normal form approach to the one-dimensional periodic cubic nonlinear Schrödinger equation in almost critical Fourier-Lebesgue spaces
• A remark on norm inflation for nonlinear Schrödinger equations
• A remark on norm inflation with general initial data for the cubic nonlinear Schrödinger equations in negative Sobolev spaces
• Stochastic nonlinear Schrödinger equation with almost space-time white noise
• Quasi-invariant Gaussian measures for the cubic fourth order nonlinear Schrödinger equation in negative Sobolev spaces
• Well-posedness for the NLS hierarchy
• Sobolev norms of $L^2$-solutions to NLS
• Invariant Gibbs dynamics for the two-dimensional Zakharov-Yukawa system
• Solitons and Gibbs measures for nonlinear Schroedinger equations