A new two-component sasa-satuma equation: large-time asymaptotics on the line

TL;DR

This paper analyzes the long-time behavior of solutions to a new two-component Sasa-Satsuma equation on the line, revealing three distinct asymptotic regions with different qualitative behaviors using Riemann-Hilbert problem techniques.

Contribution

It introduces a novel two-component Sasa-Satsuma equation and derives its large-time asymptotics via spectral analysis and nonlinear steepest descent methods.

Findings

Identification of three main asymptotic regions with distinct behaviors

Asymptotics in the central region described by a coupled Painleve II equation

Solution characterized by a fourth-order matrix Riemann-Hilbert problem

Abstract

We consider the initial value problem for a new two-component Sasa-Satsuma equation associated with the fourth-order Lax pair with decaying initial data on the line. By utilizing the spectral analysis, the solution of the new two-component Sasa-Satsuma system is transformed into the solution of a fourth-order matrix Riemann-Hilbert problem. Then the long-time asymptotics of the solution is obtained by means of the nonlinear steepest descent method of Deift and Zhou for oscillatory Riemann-Hilbert problems. We show that there are three main regions in the half-plane, where the asymptotics has qualitatively different forms: a left fast decaying sector, a central Painleve sector where the asymptotics is described in terms of the solution of a new coupled Painleve II equation which is related to a fourth-order matrix Riemann-Hilbert problem, and a right slowly decaying oscillatory sector.