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

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.