Noncommutative reduction of the nonlinear Schr\"{o}dinger equation on Lie groups

Abstract

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 Schr\"{o}dinger equation on a Lie group in curved space with local cubic nonlinearity. General formalism is illustrated by the example of noncommutative reduction of the nonstationary nonlinear Schr\"{o}dinger equation on the motion group $E(2)$ of the two-dimensional plane $\mathbb{R}^{2}$. In the particular case, we come to the usual ($1+1$) dimensional nonlinear Schr\"{o}dinger equation with the soliton solution. Another example provides the noncommutative reduction of the stationary multidimensional nonlinear Schr\"{o}dinger equation on the four-dimensional exponential solvable group.