Quadratic-Argument Approach to Nonlinear Schrödinger Equation and Coupled Ones

Abstract

The two-dimensional cubic nonlinear Schrödinger equation is used to describe the propagation of an intense laser beam through a medium with Kerr nonlinearity. The coupled two-dimensional cubic nonlinear Schrödinger equations are used to describe the interaction of electromagnetic waves with different polarizations in nonlinear optics. Mathematically, they are fundamental nonlinear partial differential equations of elliptic type. In this paper, we solve the above equations by imposing a quadratic condition on the related argument functions and using their symmetry transformations. More complete families of exact solutions of such type are obtained. Many of them are the periodic, quasi-periodic, aperiodic and singular solutions that may have practical significance.

Quadratic-Argument Approach to the Davey-Stewartson Equations

“…Clarkson and Hood [5] (1994) obtained certain symmetry reductions of the equations to ordinary differential equations with no intervening steps and provided new exact solutions which are not obtainable by the Lie group approach. Guil and Manas We observed in [17] that the argument functions of all the solutions of the twodimensional cubic nonlinear Schrödinger equation in [15] are quadratic in spacial variables. This motivated us [17] to introduce a quadratic-argument approach to study exact solutions of the two-dimensional cubic nonlinear Schrödinger equation and the coupled twodimensional cubic nonlinear Schrödinger equations modulo the known symmetry transformations.…”

“…Guil and Manas We observed in [17] that the argument functions of all the solutions of the twodimensional cubic nonlinear Schrödinger equation in [15] are quadratic in spacial variables. This motivated us [17] to introduce a quadratic-argument approach to study exact solutions of the two-dimensional cubic nonlinear Schrödinger equation and the coupled twodimensional cubic nonlinear Schrödinger equations modulo the known symmetry transformations. Indeed, our solution sets are most complete among the ones whose argument functions are quadratic in spacial variables.…”