Closed-form solutions of the nonlinear Schr\"odinger equation with arbitrary dispersion and potential

TL;DR

This paper derives exact closed-form solutions for the general nonlinear Schrödinger equation with arbitrary dispersion and potential, using a novel combination of analytical methods, applicable across physics fields.

Contribution

It introduces a new approach combining functional constraints and generalized separation of variables to solve a broad class of nonlinear Schrödinger equations.

Findings

Exact solutions expressed in quadratures or elementary functions.

Reduction to simpler ordinary differential equations.

Solutions useful for testing numerical and analytical methods.

Abstract

For the first time, the general nonlinear Schr\"odinger equation is investigated, in which the chromatic dispersion and potential are specified by two arbitrary functions. The equation in question is a natural generalization of a wide class of related nonlinear partial differential equations that are often used in various areas of theoretical physics, including nonlinear optics, superconductivity and plasma physics. To construct exact solutions, a combination of the method of functional constraints and methods of generalized separation of variables is used. Exact closed-form solutions of the general nonlinear Schr\"odinger equation, which are expressed in quadratures or elementary functions, are found. One-dimensional non-symmetry reductions are described, which lead the considered nonlinear partial differential equation to a simpler ordinary differential equation or a system of such equations. The exact solutions obtained in this work can be used as test problems intended to assess the accuracy of numerical and approximate analytical methods for integrating nonlinear equations of mathematical physics.