Department of Mathematics,

Faculty of Science and Letters,

Istanbul Technical University,

34469 Maslak/Istanbul,

TURKEY

E-mail: ozemir@itu.edu.tr, ozemirc@yahoo.com

**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.