Paper detail

Noncommutative reduction of the nonlinear Schrödinger 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 Schrödinger 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 Schrödinger 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 Schrödinger equation with the soliton solution. Another example provides the noncommutative reduction of the stationary multidimensional nonlinear Schrödinger equation on the four-dimensional exponential solvable group.