Non-point symmetry reduction method of partial differential equations

2017/12/21 by Ivan Tsyfra, Tsyfra, Ivan, Tomasz Czyżycki +1
Physics and Astronomy · #Nonlinear Waves and Solitons #Advanced Fiber Laser Technologies #Nonlinear Photonic Systems

paper · pdf · doi:10.48550/arxiv.1712.08106

Abstract

We study the symmetry reduction of nonlinear partial differential equations with two independent variables. We propose new ansätze reducing nonlinear evolution equations to system of ordinary differential equations. The ansätze are constructed by using operators of non-point classical and conditional symmetry. Then we find solution to nonlinear heat equation which can not be obtained in the framework of the classical Lie approach. By using operators of Lie--Bäcklund symmetries we construct the solutions of nonlinear hyperbolic equations depending on arbitrary smooth function of one variable too. We show that the method can be applied to nonevolutionary partial differential equations.

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• On the symmetry approach to reduction of partial differential equations
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• The natural extension to PDEs of Lie's reduction of order algorithm for ODEs