Exact solutions and self-similar symmetries of a nonlocal nonlinear Schrödinger equation

Abstract

An analytical consideration of a nonlocal nonlinear Schrodinger equation with distributed coefficients is presented. The analysis is twofold: first, exact solutions of the nonlocal equation with constant coefficients are found along with the appropriate conditions for their existence, and second, using a similarity transformation the distributed equation, i.e. the equations with varying coefficients, is reduced to the constant coefficients case; the solutions of the latter will be utilized in the construction of the solutions for the general case. This similarity transformation is also physically relevant as it allows for the physical parameters of the solution (amplitude, width, centre and phase) to vary in a specific way so that the constant coefficient system is a natural similarity reduction of the distributed case. Certain compatibility conditions needed for the consistency of this reduction are also deduced.