Initial-boundary value problems for integrable evolution equations with $3 \times 3$ Lax pairs

TL;DR

This paper extends the Fokas method for analyzing initial-boundary value problems to integrable equations with $3 imes 3$ Lax pairs, broadening the applicability of boundary analysis techniques beyond the well-studied $2 imes 2$ cases.

Contribution

The paper develops a new framework for analyzing boundary value problems for integrable equations with $3 imes 3$ Lax pairs, expanding the Fokas method to higher-dimensional Lax matrices.

Findings

Extended the inverse scattering transform to $3 imes 3$ Lax pairs.

Provided a systematic approach for boundary value problems with higher-dimensional Lax pairs.

Demonstrated the applicability of the method to new classes of integrable equations.

Abstract

We present an approach for analyzing initial-boundary value problems for integrable equations whose Lax pairs involve $3 \times 3$ matrices. Whereas initial value problems for integrable equations can be analyzed by means of the classical Inverse Scattering Transform (IST), the presence of a boundary presents new challenges. Over the last fifteen years, an extension of the IST formalism developed by Fokas and his collaborators has been successful in analyzing boundary value problems for several of the most important integrable equations with $2 \times 2$ Lax pairs, such as the Korteweg-de Vries, the nonlinear Schr\"odinger, and the sine-Gordon equations. In this paper, we extend these ideas to the case of equations with Lax pairs involving $3 \times 3$ matrices.