Initial Value Problems for Integrable Systems on a Semi-Strip

TL;DR

This paper rigorously analyzes initial value problems for integrable systems on a semi-strip, focusing on boundary conditions and solutions for the nonlinear Schrödinger and N-wave equations, with new results for matrix solutions and boundary conditions.

Contribution

It provides new rigorous results on boundary value problems for integrable systems, including matrix solutions of the nonlinear Schrödinger equation and specific cases of the N-wave equation.

Findings

Unique determination of solutions by initial conditions

New results for matrix solutions of the nonlinear Schrödinger equation

Analysis of boundary conditions for N-wave equation

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 Schr\"odinger equation with quasi-analytic boundary conditions is dealt with. (The result is new even for a scalar nonlinear Schr\"odinger equation.) Next, a special case of the nonlinear optics ($N$-wave) equation is considered.