Geometric Optics and Long Range Scattering¶for One-Dimensional Nonlinear Schrödinger Equations

Abstract

With the methods of geometric optics used in [2], we provide a new proof of some results of [10], to construct modified wave operators for the one-dimensional cubic Schrodinger equation. We improve the rate of convergence of the nonlinear solution towards the simplified evolution, and get better control of the loss of regularity in Sobolev spaces. In particular, using the results of [9], we deduce the existence of a modified scattering operator with small data in some Sobolev spaces. We show that in terms of geometric optics, this gives rise to a “random phase shift” at a caustic.