A Two-Component Sasa–Satsuma Equation: Large-Time Asymptotics on the Line

Abstract

We consider the initial value problem for a two-component Sasa–Satsuma equation associated with a \(4\times 4\) Lax pair with decaying initial data on the line. By utilizing the spectral analysis, the solution of the two-component Sasa–Satsuma system is transformed into the solution of a \(4\times 4\) 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 \(-\infty<x<\infty \) , \(t>0\) , where the asymptotics has qualitatively different forms: a left fast decaying sector, a central Painlevé sector where the asymptotics is described in terms of the solution to a system of coupled Painlevé II equations, which is related to a \(4\times 4\) matrix Riemann–Hilbert problem, and a right slowly decaying oscillatory sector.