The scattering problem for a noncommutative nonlinear Schrödinger equation

Abstract

We investigate scattering properties of a Moyal deformed version of the nonlinear Schrödinger equation in an even number of space dimensions. With rather weak conditions on the degree of nonlinearity, the Cauchy problem for general initial data has a unique globally defined solution, and also has soliton solutions if the interaction potential is suitably chosen. We demonstrate how to set up a scattering framework for equations of this type, including appropriate decay estimates of the free time evolution and the construction of wave operators defined for small scattering data in the general case and for arbitrary scattering data in the rotationally symmetric case.

Published version

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• On nonlocal reductions of the multi-component nonlinear Schrodinger equation on symmetric spaces
• Multicomponent Fokas-Lenells equations on Hermitian symmetric spaces
• NLS-type equations from quadratic pencil of Lax operators: negative flows
• Linear hyperbolic PDEs with non-commutative time
• On soliton solutions and soliton interactions of Kulish-Sklyanin and Hirota-Ohta systems
• $G$-Strands on symmetric spaces
• Structure of Noncommutative Solitons: Existence and Spectral Theory
• Twisted hierarchies associated with the generalized sine-Gordon equation
• Dynamics of Noncommutative Solitons II: Spectral Theory, Dispersive Estimates and Stability
• Integrable systems on symmetric spaces from a quadratic pencil of Lax operators
• High energy bounds on wave operators