Group-Invariant Solutions of (2+1)-Dimensional Cubic Schr\"odinger Equation

Abstract:
Group invariant solutions for (2+1)-dimensional cubic Schr\"odinger equation (CSE)are investigated. The symmetry algebra of CSE is already known. All the solutions invariant under symmetry groups corresponding to non-trivial two and three-dimensional subalgebras of this algebra are investigated. By the use of these subalgebras in symmetry reduction, reduction of the equation to ODEs or algebraic equations are obtained. Exact solutions in terms of trigonometric functions,
elliptic functions and Painlev\'e transcendent functions are found for the reduced equations which have the Painlev\'e property.