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

An analytical study, strongly aided by computer algebra packages diffgrob2 by Mansfield and rif by Reid, is made of the 3 + 1-coupled nonlinear Schrod inger (CNLS) system i Psi(t) + del(2)Psi + (\Psi\(2) + \Phi\(2)) Psi = 0, i Phi(t) + del(2)Phi + (\Psi\(2) + \Phi\(2)) Phi = 0. This system describes transverse effects in nonlinear optical systems. It also arises in the stud y of the transmission of coupled wave packets and "optical solitons", in no nlinear optical fibres. First we apply Lie's method for calculating the classical Lie algebra of ve ctor fields generating symmetries that leave invariant the set of solutions of the CNLS system. The large linear classical determining system of PDE f or the Lie algebra is automatically generated and reduced to a standard for m by the rif algorithm, then solved, yielding a 15-dimensional classical Li e 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 ide ntifying nonclassical vector fields which leave invariant the joint solutio n set of the CNLS system and a certain additional system, called the invari ant surface condition. In the generic case the system of determining equati ons has 856 PDE, is nonlinear and considerably more complicated than the li near classical system of determining equations whose solutions it possesses as a subset. Very few calculations of this magnitude have been attempted d ue to the necessity to treat cases, expression explosion and until recent t imes the dearth of mathematically rigorous algorithms for nonlinear systems . The application of packages diffgrob2 and rif leads to the explicit solutio n of the nonclassical determining system in eleven cases. Action of the cla ssical group on the nonclassical vector fields considerably simplifies one of these cases. We identify the reduced form of the CNLS system in each cas e. Many of the cases yield new results which apply equally to a generalized coupled nonlinear Schrodinger system in which \Psi\(2) + \Phi\(2) may be r eplaced by an arbitrary function of \Psi\(2) + \Phi\(2). Coupling matrices in sl(2, C) feature prominently in this family of reductions. (C) 1998 Else vier Science B.V.