The two-dimensional cubic nonlinear Schrodinger equation is used to describe the propagation of an intense laser beam through a medium with Kerr nonlinearity. The coupled two-dimensional cubic nonlinear Schrodinger 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.
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