Asymptotic solutions of the nonlocal nonlinear Schrödinger equation in the limit of small dispersion

Abstract

A method is developed for solving the initial value problem of the nonlocal nonlinear Schrödinger (NLS) equation in the limit of small dispersion. Whitham's modulation theory is used to characterize the main feature of the solution in terms of the single-phase periodic solution of the nonlocal NLS equation with the slowly varying wave parameters. The modulation equations for these parameters are derived by averaging the local conservation laws. A novel feature of the modulation equations is that they can be decoupled into the form of the integrable Hopf equation. An explicit example of the solution is exhibited for a step initial condition.