Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 6 (2010), 046, 17 pages      arXiv:0903.1493      https://doi.org/10.3842/SIGMA.2010.046
Contribution to the Special Issue “Noncommutative Spaces and Fields”

The Scattering Problem for a Noncommutative Nonlinear Schrödinger Equation

Bergfinnur Durhuus a and Victor Gayral b
a) Department of Mathematics, Copenhagen University, Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark
b) Laboratoire de Mathématiques, Université de Reims Champagne-Ardenne, Moulin de la Housse - BP 1039 51687 Reims cedex 2, France

Received March 03, 2010, in final form May 20, 2010; Published online June 03, 2010

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 solitary wave 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.

Key words: noncommutative geometry; nonlinear wave equations; scattering theory; Jacobi polynomials.

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