Application of the Nonlinear Steepest Descent Method to the Coupled Sasa-Satsuma Equation

Abstract

We use spectral analysis to reduce Cauchy problem for the coupled Sasa-Satsuma equation to a 5 × 5 matrix Riemann-Hilbert problem. The upper and lower triangular factorisations of the jump matrix and a decomposition of the vector-valued spectral function are given. Applying various transformations related to the Riemann-Hilbert problems and suitable decompositions of the jump contours and the nonlinear steepest descent method, we establish the long-time asymptotics of the problem.

Reverse-time type nonlocal Sasa–Satsuma equation and its soliton solutions

“…The symmetry (42) indicates that ( ) det Y + at (x, t, λ * ) equates to ( ) det Yat (x, t, λ). Thus, based on (43) and N l l…”

“…This indicates that one only needs to solve v j , (1 j N 1 ), such that all vectors v j and vj can be obtained. Thus by taking the x, t-derivatives of the first expression in equation (43) and integrating it, we have…”

“…Based on the nonlinear steepest descent approach, the long-time asymptotic behavior of solutions has been reported in [40]. Even though the CSS equation has been studied and generated much important work [42][43][44], to the best of our knowledge, its solutions with special initial solutions v(x, 0) = u * (x, 0) have not received much attention.…”