Long-time asymptotics for a fourth-order dispersive nonlinear Schrödinger equation with nonzero boundary conditions

Abstract

In this work, we consider the long-time asymptotics of the Cauchy problem for a fourth-order dispersive nonlinear Schrödinger equation with nonzero boundary conditions at infinity. Firstly, in order to construct the basic Riemann-Hilbert problem associated with nonzero boundary conditions, we analyze direct scattering problem. The nonlinear steepest descent method is employed to transform the matrix Riemann-Hilbert problem into a solvable model. Furthermore, the g-function mechanism is applied to effectively eliminate the exponential growth in the jump matrix. We obtain the long-time asymptotic behavior in the modulated elliptic wave region and the plane wave region for the fourth-order dispersive nonlinear Schrödinger equation. Finally, we also provide an analysis of the modulation instability of the initial plane wave.