A pseudoconformal compactification of the nonlinear Schrödinger equation and applications

Abstract

We interpret the lens transformation (a variant of the pseudoconformal transformation) as a pseudoconformal compactification of spacetime, which converts the nonlinear Schrödinger equation (NLS) without potential with a nonlinear Schrödinger equation with attractive harmonic potential. We then discuss how several existing results about NLS can be placed in this compactified setting, thus offering a new perspective to view this theory.

17 pages, no figures, to appear, New York J. Math. This is the final version

References in corpus (4)

• Critical nonlinear Schroedinger equations with and without harmonic potential
• The mass-critical nonlinear Schrödinger equation with radial data in dimensions three and higher
• The nonlinear Schrödinger equation with combined power-type nonlinearities
• Similarity Solutions and Collapse in the Attractive Gross-Pitaevskii Equation

Cited by in corpus (15)

• The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher
• Compactification for Asymptotically Autonomous Dynamical Systems: Theory, Applications and Invariant Manifolds
• Derivation of the Time Dependent Two Dimensional Focusing NLS Equation
• Fermi-Pasta-Ulam phenomena and persistent breathers in the harmonic trap
• Energy level splitting for weakly interacting bosons in a harmonic trap
• Some qualitative studies of the focusing inhomogeneous Gross-Pitaevskii equation
• On the continuous resonant equation for NLS: I. Deterministic analysis
• Local maximizers of adjoint Fourier restriction estimates for the cone, paraboloid and sphere
• Two Dimensional NLS Equation With Random Radial Data
• On scattering for the quintic defocusing nonlinear Schrödinger equation on \R \times \T^2
• Global existence and uniqueness results for weak solutions of the focusing mass-critical non-linear Schrödinger equation
• Large time behavior in nonlinear Schrodinger equation with time dependent potential
• The Hartree equation with a constant magnetic field: Well-posedness theory
• Asymptotic behavior of nonlinear Schroedinger Systems with Linear Coupling
• On interpolation of cocompact imbeddnings