Multiphase solutions and their reductions for a nonlocal nonlinear Schrödinger equation with focusing nonlinearity

Abstract

AbstractA nonlocal nonlinear Schrödinger equation with focusing nonlinearity is considered, which has been derived as a continuum limit of the Calogero–Sutherland model in an integrable classical dynamical system. The equation is shown to stem from the compatibility conditions of a system of linear partial differential equations (PDEs), assuring its complete integrability. We construct a nonsingular N‐phase solution (N: positive integer) of the equation by means of a direct method. The features of the one‐ and two‐phase solutions are investigated in comparison with the corresponding solutions of the defocusing version of the equation. We also provide an alternative representation of the N‐phase solution in terms of solutions of a system of nonlinear algebraic equations. Furthermore, the eigenvalue problem associated with the N‐phase solution is discussed briefly with some exact results. Subsequently, we demonstrate that the N‐soliton solution can be obtained simply by taking the long‐wave limit of the N‐phase solution. The similar limiting procedure gives an alternative representation of the N‐soliton solution as well as the exact results related to the corresponding eigenvalue problem.