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arXiv · 1904.01854

Symmetries and reductions of integrable nonlocal partial differential equations

Linyu Peng
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Abstract

In this paper, symmetry analysis is extended to study nonlocal differential equations, in particular two integrable nonlocal equations, the nonlocal nonlinear Schr\"odinger equation and the nonlocal modified Korteweg--de Vries equation. Lie point symmetries are obtained based on a general theory 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. At the end, we also define complex transformations to connect nonlocal differential equations and differential-difference equations.

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Linyu Peng. 2019-04-03. Symmetries and reductions of integrable nonlocal partial differential equations. https://doi.org/10.3390/sym11070884

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