Exact Analytical Solutions for Nonautonomic Nonlinear Klein-Fock-Gordon Equation

  • Eron L. Aero
  • A. N. BulyginEmail author
  • Yu. V. Pavlov
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 87)


We develop methods of construction of functionally invariant solutions U(xyzt) for the nonlinear nonautonomic Klein-Fock-Gordon equation. The solutions U(xyzt) are found in the form of an arbitrary function, which depends on one, \(\tau (x,y,z,t)\), or two, \(\alpha (x,y,z,t)\), \(\beta (x,y,z,t)\) specially constructed functions. The functions are called ansatzes. The ansatzes \((\tau , \alpha , \beta )\) are defined as solutions of the special equations (algebraic or mixed type—algebraic and partial differential equations). The equations for defining of the ansatzes contain arbitrary functions, depending on \((\tau , \alpha , \beta )\). The offered methods allow to find the solution U(xyzt) for particular, but wide class of the nonautonomic nonlinear Klein-Fock-Gordon equations. The methods are illustrated by examples of finding exact analytical solutions of the nonautonomic Liouville equation.



This work was supported by Russian Foundation for Basic Research, grants 16-01-00068-a and 17-01-00230-a.


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© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Institute for Problems in Mechanical Engineering of Russian Academy of SciencesSaint PetersburgRussia

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