Exactly solvable models of nonlinear extensions of the Schr\"odinger equation

Abstract

A method is presented to construct exactly solvable nonlinear extensions of the Schr\"odinger equation. The method explores a correspondence which can be established under certain conditions between exactly solvable ordinary Schr\"odinger equations and exactly solvable nonlinear theories. We provide several examples illustrating the method. We rederive well-known soliton solutions and find new exactly solvable nonlinear theories in various space dimensions which, to the best of our knowledge, have not yet been discussed in literature. Our method can be used to construct further nonlinear theories and generalized to relativistic soliton theories, and may have many applications.