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

SIGMA 12 (2016), 001, 17 pages      arXiv:1405.3500      https://doi.org/10.3842/SIGMA.2016.001

Initial Value Problems for Integrable Systems on a Semi-Strip

Alexander L. Sakhnovich
Vienna University of Technology, Institute of Analysis and Scientific Computing, Wiedner Hauptstr. 8, A-1040 Vienna, Austria

Received September 01, 2015, in final form December 28, 2015; Published online January 03, 2016

Abstract
Two important cases, where boundary conditions and solutions of the well-known integrable equations on a semi-strip are uniquely determined by the initial conditions, are rigorously studied in detail. First, the case of rectangular matrix solutions of the defocusing nonlinear Schrödinger equation with quasi-analytic boundary conditions is dealt with. (The result is new even for a scalar nonlinear Schrödinger equation.) Next, a special case of the nonlinear optics ($N$-wave) equation is considered.

Key words: Weyl-Titchmarsh function; initial condition; quasi-analytic functions; system on a semi-strip; nonlinear Schrödinger equation; nonlinear optics equation.

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