Generalized and functional separable solutions to nonlinear delay Klein–Gordon equations

• Published in: Communications in nonlinear science & numerical simulation 2014-08, Vol.19 (8), p.2676-2689
• Main Authors: Polyanin, Andrei D., Zhurov, Alexei I.
• Format: Article
• Language: English
• Subjects: Approximation; Delay; Delay partial differential equations; Exact solutions; Functional separable solutions; Generalized separable solutions; Klein-Gordon equation; Mathematical analysis; Mathematical models; Nonlinear Klein–Gordon equations; Nonlinearity; Texts
• ISSN: 1007-5704; 1878-7274
• DOI: 10.1016/j.cnsns.2013.12.021

•Non-linear delay Klein–Gordon equations are considered.•New generalized and functional separable solutions are presented.•Most equations contain arbitrary functions of one argument.•Some equations contain an arbitrary function of two arguments.•Some solutions involve infinitely many free parameters. We describe a number of generalized separable, functional separable, and some other exact solutions to nonlinear delay Klein–Gordon equations of the formutt=kuxx+F(u,w),where u=u(x,t) and w=u(x,t-τ), with τ denoting the delay time. The generalized separable solutions are sought in the form u=∑n=1NΦn(x)Ψn(t), where the functions Φn(x) and Ψn(t) are to be determined subsequently. Most of the equations considered contain one or two arbitrary functions of a single argument or one arbitrary function of two arguments of special form. We present a substantial number of new exact solutions, including periodic and antiperiodic ones, as well as composite solutions resulting from a nonlinear superposition of generalized separable and traveling wave solutions. All solutions involve free parameters (in some cases, infinitely many) and so can be suitable for solving certain problems and testing approximate analytical and numerical methods for nonlinear delay PDEs.