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 nonl...

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Bibliographic Details
Published in:arXiv.org 2022-08
Main Authors: Breev, A I, Shapovalov, A V, Gitman, D M
Format: Article
Language:English
Subjects:
Independent variables
Lie groups
Mathematical analysis
Nonlinear equations
Nonlinearity
Partial differential equations
Reduction
Schrodinger equation
Solitary waves
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Summary: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.
ISSN:2331-8422