A family of nonlinear Schrödinger equations admitting q-plane wave solutions

Abstract

Nonlinear Schrödinger equations with power-law nonlinearities have attracted considerable attention recently. Two previous proposals for these types of equations, corresponding respectively to the Gross– Pitaievsky equation and to the one associated with nonextensive statistical mechanics, are here unified into a single, parameterized family of nonlinear Schrödinger equations. Power-law nonlinear terms characterized by exponents depending on a real index q, typical of nonextensive statistical mechanics, are considered in such a way that the Gross–Pitaievsky equation is recovered in the limit q → 1. A classical field theory shows that, due to these nonlinearities, an extra field ( x,t) (besides the usual one ( x,t)) must be introduced for consistency. The new field can be identified with ∗( x,t) only when q → 1. For q = 1 one has a pair of coupled nonlinear wave equations governing the joint evolution of the complex valued fields ( x,t) and ( x,t). These equations reduce to the usual pair of complex-conjugate ones only in the q → 1 limit. Interestingly, the nonlinear equations obeyed by ( x,t) and ( x,t) exhibit a common, soliton-like, traveling solution, which is expressible in terms of the q-exponential function that naturally emerges within nonextensive statistical mechanics.