Noncommutative Reduction of Nonlinear Schrodinger Equation on Lie Groups

We propose a new approach that allows one to reduce nonlinear equations on Lie groups to equations with a fewer number of independent variables for finding particular solutions of the nonlinear equations. The main idea is to apply the method of noncommutative integration to the linear part of a nonlinear equation, which allows one to find bases in the space of solutions of linear partial differential equations with a set of noncommuting symmetry operators. The approach is implemented for the generalized nonlinear Schrodinger equation on a Lie group in curved space with local cubic nonlinearity. General formalism is illustrated by the example of the noncommutative reduction of the nonstationary nonlinear Schrodinger equation on the motion group E (2) of the two-dimensional plane R-2. In this particular case, we come to the usual (1 + 1)-dimensional nonlinear Schrodinger equation with the soliton solution. Another example provides the noncommutative reduction of the stationary multidimensional nonlinear Schrodinger equation on the four-dimensional exponential solvable group.