Nonlinear wave-wave interaction and stability criterion for parametrically coupled nonlinear Schrodinger equations

A rigorous mathematical reduction of the procedure widely used for studying a class of the nonlinear problems with perturbations, namely the method of the multiple scales, is used. A profound analysis, which provides an approach for deriving a coupled nonlinear Schrodinger equations. The investigation has been achieved by perturbing the nonlinear dynamical system about the linear dynamical problem. Modulated wavetrains are described to all orders of approximation. Moreover, we extend our approach to deal with equations having periodic terms. Two types of simultaneous nonlinear Schrodinger equations are derived. One type is valid at the non-parametric system and the second type represents a modification for the first type which is governed the non-resonance case. Two parametric coupled nonlinear Schrodeinger equations are derived to govern the second-sub-harmonic resonance. In addition other two coupled equations are found for the third-sub-harmonic resonance case. These systems of equations control the stability behavior at the parametric resonance cases. The stability criteria for the several types of coupled nonlinear Schrodinger equations are studied. These criteria are achieved by a temporal periodic perturbation.