The unified transform method for the Sasa-Satsuma equation on the interval

TL;DR

This paper develops a Riemann-Hilbert problem framework for solving the Sasa-Satsuma equation on a finite interval, explicitly relating spectral functions to initial and boundary data, and analyzing boundary value determination.

Contribution

It introduces a Riemann-Hilbert formalism for the Sasa-Satsuma equation on finite intervals, explicitly characterizing boundary data via the global relation.

Findings

Expresses solution in terms of a 3x3 Riemann-Hilbert problem.

Defines spectral functions based on initial and boundary data.

Analyzes the global relation to determine unknown boundary values.

Abstract

We present a Riemann-Hilbert problem formalism for the initial-boundary value problem for the Sasa-Satsuma(SS) equation on the finite interval. Assume that the solution existes, we show that this solution can be expressed in terms of the solution of a $3\times 3$ Riemann-Hilbert problem. The relevant jump matrices are explicitly given in terms of the three matrix-value spectral functions $s(k)$, $S(k)$ and $S_L(k)$, which in turn are defined in terms of the initial values, boundary values at $x=0$ and boundary values at $x=L$, respectively. However, for a well-posed problem, only part of the boundary values can be prescribed, the remaining boundary data cannot be independently specified, but are determined by the so-called global relation. Here, we analyze the global relation to characterize the unknown boundary values in terms of the given initial and boundary data.