Nonclassical reductions of a 3+1-cubic nonlinear Schrödinger system

Abstract

An analytical study, strongly aided by computer algebra packages diffgrob2 by Mansfield and rif by Reid, is made of the 3+1-coupled nonlinear Schrödinger (CNLS) system i ψ l +▿ 2 ψ + (| ψ| 2 + | Φ| 2) ψ = 0, i Φ l +▿ 2 Φ+(| ψ| 2 + | Φ| 2) Φ = 0. This system describes transverse effects in nonlinear optical systems. It also arises in the study of the transmission of coupled wave packets and “optical solitons”, in nonlinear optical fibres. First we apply Lie's method for calculating the classical Lie algebra of vector fields generating symmetries that leave invariant the set of solutions of the CNLS system. The large linear classical determining system of PDE for the Lie algebra is automatically generated and reduced to a standard form by the rif algorithm, then solved, yielding a 15-dimensional classical Lie invariance algebra. A generalization of Lie's classical method, called the nonclassical method of Bluman and Cole, is applied to the CNLS system. This method involves identifying nonclassical vector fields which leave invariant the joint solution set of the CNLS system and a certain additional system, called the invariant surface condition. In the generic case the system of determining equations has 856 PDE, is nonlinear and considerably more complicated than the linear classical system of determining equations whose solutions it possesses as a subset. Very few calculations of this magnitude have been attempted due to the necessity to treat cases, expression explosion and until recent times the dearth of mathematically rigorous algorithms for nonlinear systems. The application of packages diffgrob2 and rif leads to the explicit solution of the nonclassical determining system in eleven cases. Action of the classical group on the nonclassical vector fields considerably simplifies one of these cases. We identify the reduced form of the CNLS system in each case. Many of the cases yield new results which apply equally to a generalized coupled nonlinear Schrödinger system in which | ψ| 2 + | Φ| 2 may be replaced by an arbitrary function of | ψ| 2 + | Φ| 2. Coupling matrices in sl(2, C ) feature prominently in this family of reductions.