The nonlinear steepest descent approach for long time behavior of the two-component coupled Sasa-Satsuma equation with a $5\times 5$ Lax pair

TL;DR

This paper applies the nonlinear steepest descent method to analyze the long-time behavior of the coupled Sasa-Satsuma equation, modeled via a 5x5 Lax pair, relevant for optical pulse propagation in birefringent fibers.

Contribution

It formulates a Riemann-Hilbert problem based on a 5x5 spectral problem and uses the Deift-Zhou method to study the equation's long-term dynamics.

Findings

Long-time asymptotics derived for the coupled Sasa-Satsuma equation.

Representation of solutions via Riemann-Hilbert problem.

Application of steepest descent method to a 5x5 Lax pair.

Abstract

Under investigation in this work is the coupled Sasa-Satsuma equation, which can describe the propagations of two optical pulse envelopes in birefringent fibers. The Riemann-Hilbert problem for the equation is formulated on the basis of the corresponding $5\times5$ matrix spectral problem, which allows us to present a suitable representation for the solution of the equation. Then the Deift-Zhou steepest descent method is used to analyze the long time behavior of the coupled Sasa-Satsuma equation.