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This paper has been accepted for publication in Contemporary Mathematics. To download a preprint of this paper just click here.

This material is based upon work supported by the National Science Foundation under Grant Nos. DMS-0807653 and DMS-1206131. Any opinions, findings and conclusions or recomendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation (NSF).

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The Semiclassical Modified Nonlinear Schrödinger Equation II: Asymptotic Analysis of the Cauchy Problem. The Elliptic Region for Transsonic Initial Data.

Jeffery C. DiFranco and Peter D. Miller
Department of Mathematics, Seattle University
Department of Mathematics, University of Michigan, Ann Arbor


We begin a study of a multi-parameter family of Cauchy initial-value problems for the modified nonlinear Schrödinger equation, analyzing the solution in the semiclassical limit. We use the inverse scattering transform for this equation, along with the steepest descent method of Deift and Zhou. The initial conditions are selected both to allow all relevant scattering data to be calculated without approximation and also to place the governing equation in a transsonic state in which the quantum fluid dynamical system formally approximating it is of hyperbolic type for some x and of elliptic type for other x. Our main result is a global approximation theorem valid in a maximal space-time region connected to the elliptic part of the initial data.