Families of nonlinear Schrödinger equations in general form with exact solutions

Abstract

The family of nonlinear Schrödinger equations in its the general form is considered. These nonlinear partial differential equations depend on arbitrary functions, and the Cauchy problems for them are not solved by the inverse scattering transform in the general case. However, we demonstrate that a few families of nonlinear Schrödinger equations admit the generalized traveling wave solutions. Our approach relies on functions satisfying first- and second-order differential equations. In fact, these functions can be considered as the results of experimental measurements of pulses in a nonlinear medium. Thus, the proposed approach can be seen as an attempt to define a mathematical model based on measurement results. In this paper we use solutions of first-order ordinary differential equations with known forms. This idea allows us to find a condition on arbitrary functions. In this case, one constraint is imposed on arbitrary functions that determines the family of nonlinear partial differential equations under consideration. Four families of new nonlinear Schrödinger equations are presented, which admit generalized traveling wave solutions expressed in terms of elementary functions.