# Journal of Nonlinear Mathematical Physics

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

c(x)utt+d(x)ut=[a(x)ux]x+b(x)ux+p(x)f(u),
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.

Open Access

Journal
Journal of Nonlinear Mathematical Physics
Volume-Issue
27 - 2
Pages
227 - 242
Publication Date
2020/01/27
ISSN (Online)
1776-0852
ISSN (Print)
1402-9251
DOI
10.1080/14029251.2020.1700633How to use a DOI?
Open Access

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  -