Singular point analysis and integrals of motion for coupled nonlinear Schrödinger equations

Abstract

For a system of two coupled nonlinear Schrödinger equations and the corresponding Madelung fluid equations we calculate the similarity transformation and reductions to ordinary differential equations. The symmetries can be used to classify several types of solutions. Besides a system of coupled Painlevé II equations we obtain a two dimensional quartic hamiltonian system. Using Noether’s theorem and performing a Painlevé-test we identify those parameters which lead to regular motion on tori. Using the results of the P-test we obtain heretofore undiscovered integrals of motion for the Madelung fluid by direct calculations. Investigating the dynamics of the system for parameters that are different from those obtained by the Painlevé-test we calculate numerically the surface of section (KAM-tori) and study the transition into chaotic behaviour.