A new approach to integrable evolution equations on the circle

TL;DR

This paper introduces a novel method for solving integrable evolution equations on the circle by employing a unified transform and Riemann-Hilbert problem formulation, extending inverse scattering techniques to periodic settings.

Contribution

It develops a new approach for initial value problems on the circle using the unified transform, linking initial data to Riemann-Hilbert problems for integrable equations.

Findings

Solution expressed via Riemann-Hilbert problem

Applicable to the nonlinear Schrödinger equation

Effective method for periodic initial value problems

Abstract

We propose a new approach for the solution of initial value problems for integrable evolution equations in the periodic setting based on the unified transform. Using the nonlinear Schr\"odinger equation as a model example, we show that the solution of the initial value problem on the circle can be expressed in terms of the solution of a Riemann-Hilbert problem whose formulation involves quantities which are defined in terms of the initial data alone. Our approach provides an effective solution of the problem on the circle which is conceptually analogous to the solution of the problem on the line via the inverse scattering transform.