Non-existence of certain elliptic solutions of the Cubic-nonlinear Schrödinger Equation

Abstract

For a certain class of solutions of the cubic nonlinear Schrödinger equation we prove non-existence in the generic case. In the nongeneric case we present a two-parameter set of solutions, bounded or unbounded, depending on corresponding constraints. I. INTRODUCTIONAs is well known, elliptic (travelling-wave) solutions of the non-integrable complex Ginzburg-Landau equation 1) is integrable and coincides with the cubic nonlinear Schrödinger equation (CNLSE)where, in the "fiber optics notation", z denotes the distance along the fiber, and t the (retarded) time. Besides general solutions derived by direct methods (e.g., IST, Darboux), particular solutions of the CNLSE, suitable for specific physical applications, are interesting and may suffice. In this context, methods have been proposed for seeking elliptic solutions of Eq.( 1) [2(b)]. For the CNLSE particular elliptic travelling-wave solutions exist in the form (see, e.g., [3] and references therein)Thus, in correspondence to the non-existence of elliptic solutions of Eq.( 1), the question is obvious whether elliptic solutions of Eq.( 2), more general than those represented by (3), exist. In particular, if f (z)e −iλt in (3) is replaced by f (t, z) + id(z) (without assuming travelling-wave reduction z = x − ct), we are led to Ψ(t, z) = (f (t, z) + id(z))e iφ(z) , f, φ, d ∈ R, (4) as a possible ansatz, suitable to obtain elliptic solutions of the CNLSE (2).