The unified transform method to the high-order nonlinear Schrödinger equation with periodic initial condition

Abstract

In this paper, we study the high-order nonlinear Schrödinger equation with periodic initial conditions via the unified transform method extended by Fokas and Lenells. For the high-order nonlinear Schrödinger equation, the initial value problem on the circle can be expressed in terms of the solution of a Riemann–Hilbert problem. The related jump matrix can be explicitly expressed based on the initial data alone. Furthermore, we present the explicit solution, which corresponds to a one-gap solution.