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

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 Φ(x→,t) (besides the usual one Ψ(x→,t)) must be introduced for consistency. The new field can be identified with Ψ⁎(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 Ψ(x→,t) and Φ(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 Ψ(x→,t) and Φ(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.