Quadratic-Argument Approach to Nonlinear Schrodinger Equation and Coupled Ones

The two-dimensional cubic nonlinear Schr\"{o}dinger equation is used to describe the propagation of an intense laser beam through a medium with Kerr nonlinearity. The coupled two-dimensional cubic nonlinear Schr\"{o}dinger equations are used to describe interaction of electromagnetic waves with different polarizations in nonlinear optics. 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 known interesting solutions, such soliton ones, turn out to be special cases of our solutions.