Nonlinear Schrödinger equations of general form and their exact solutions

Abstract

The wide class of nonlinear Schrödinger equation of the general form is studied. These nonlinear partial differential equations, depending on arbitrary functions, are not integrable by the inverse scattering transform but have exact solutions. The approach is proposed that makes it possible to find nonlinear Schrodinger equations of the general form that have exact solutions. This approach is that the solutions of nonlinear Schrödinger equations are expressed in a special way through the solutions of auxiliary nonlinear ordinary differential equations of the second order. In this case, one constraint is imposed on three arbitrary functions that determine the class of nonlinear partial differential equations under consideration. A number of new nonlinear Schrödinger equations are presented, which admit generalized traveling wave solutions expressed in terms of elementary and elliptic functions. The described new approach and exact solutions can be used as test problems intended to assess the accuracy of numerical and approximate analytical methods for solving complex nonlinear partial differential equations of mathematical physics.