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Symmetries and Reductions of Integrable Nonlocal Partial Differential Equations

Waseda Institute for Advanced Study, Waseda University, Tokyo 169-8050, Japan
Symmetry 2019, 11(7), 884; https://doi.org/10.3390/sym11070884
Received: 8 June 2019 / Revised: 24 June 2019 / Accepted: 2 July 2019 / Published: 5 July 2019
(This article belongs to the Special Issue Noether's Theorem and Symmetry)
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Abstract

In this paper, symmetry analysis is extended to study nonlocal differential equations. In particular, two integrable nonlocal equations are investigated, the nonlocal nonlinear Schrödinger equation and the nonlocal modified Korteweg–de Vries equation. Based on general theory, Lie point symmetries are obtained and used to reduce these equations to nonlocal and local ordinary differential equations, separately; namely, one symmetry may allow reductions to both nonlocal and local equations, depending on how the invariant variables are chosen. For the nonlocal modified Korteweg–de Vries equation, analogously to the local situation, all reduced local equations are integrable. We also define complex transformations to connect nonlocal differential equations and differential-difference equations. View Full-Text
Keywords: continuous symmetry; symmetry reduction; integrable nonlocal partial differential equations continuous symmetry; symmetry reduction; integrable nonlocal partial differential equations
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Peng, L. Symmetries and Reductions of Integrable Nonlocal Partial Differential Equations. Symmetry 2019, 11, 884.

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