Volume 27, Issue 2, January 2020, Pages 227 - 242
Symmetry reductions and new functional separable solutions of nonlinear Klein–Gordon and telegraph type equations
Authors
Alexei I. Zhurov
Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences, 101 Vernadsky Avenue, bldg 1, 119526 Moscow, Russia; Cardiff University, Heath Park, Cardiff CF14 4XY, UK,zhurovai@cardiff.ac.uk
Andrei D. Polyanin
Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences, 101 Vernadsky Avenue, bldg 1, 119526 Moscow, Russia; Bauman Moscow State Technical University, 5 Second Baumanskaya Street, 105005 Moscow, Russia; National Research Nuclear University MEPhI, 31 Kashirskoe Shosse, 115409 Moscow, Russia,polyanin@ipmnet.ru
Received 5 April 2019, Accepted 28 August 2019, Available Online 27 January 2020.
- DOI
- 10.1080/14029251.2020.1700633How to use a DOI?
- Keywords
- nonlinear Klein–Gordon equations; nonlinear telegraph equations; symmetry reductions; functional separable solutions; generalized traveling wave solutions; delay differential equations
- Abstract
The paper is concerned with different classes of nonlinear Klein–Gordon and telegraph type equations with variable coefficients
where f(u) is an arbitrary function. We seek exact solutions to these equations by the direct method of symmetry reductions using the composition of functions u = U(z) with z = φ(x,t). We show that f(u) and any four of the five functional coefficients a(x), b(x), c(x), d(x), and p(x) in such equations can be set arbitrarily, while the remaining coefficient can be expressed in terms of the others. The study investigates the properties and finds some solutions of the overdetermined system of PDEs for φ(x,t). Examples of specific equations with new exact functional separable solutions are given. In addition, the study presents some generalized traveling wave solutions to more complex, nonlinear Klein–Gordon and telegraph type equations with delay.- Copyright
- © 2020 The Authors. Published by Atlantis and Taylor & Francis
- Open Access
- This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).
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TY - JOUR AU - Alexei I. Zhurov AU - Andrei D. Polyanin PY - 2020 DA - 2020/01/27 TI - Symmetry reductions and new functional separable solutions of nonlinear Klein–Gordon and telegraph type equations JO - Journal of Nonlinear Mathematical Physics SP - 227 EP - 242 VL - 27 IS - 2 SN - 1776-0852 UR - https://doi.org/10.1080/14029251.2020.1700633 DO - 10.1080/14029251.2020.1700633 ID - Zhurov2020 ER -