Spectral Analysis and Long-Time Asymptotics of a Coupled Nonlinear Schrödinger System

Starting from the 4× 4 Lax pair associated with a new type coupled nonlinear Schrödinger system and its spectral analysis, a Riemann–Hilbert problem is constructed, and the solution of Cauchy problem of the new type coupled nonlinear Schrödinger system is transformed into the solution of the Riemann–Hilbert problem. Based on the basic Riemann–Hilbert problem, the Deift–Zhou deformations to the contour of the Riemann–Hilbert problem are considered. With the aid of the Deift–Zhou nonlinear steepest descent method, the basic Riemann–Hilbert problem is transformed into a model Riemann–Hilbert problem and we derive the leading-order asymptotics of its solution according to the asymptotic expansions of the parabolic cylinder function. Finally, we obtain the long-time asymptotic behavior of solutions to the Cauchy problem of the new type coupled nonlinear Schrödinger system.