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TMF, 2018, Volume 197, Number 3, Pages 356–370 (Mi tmf9502)  

Multidimensional nonlinear Klein–Gordon equations and rivertons

V. M. Zhuravlev

Kapitsa Technological Research Institute, Ulyanovsk State University, Ulyanovsk, Russia

Abstract: Based on solutions of a system of quasilinear first-order equations of a special kind (rivertons), we construct classes of exact solutions of multidimensional nonlinear Klein–Gordon equations. The obtained solutions are expressed in terms of the derivatives of rivertons with respect to the independent variables. As a result, the solutions are multivalued and have singularities at the branch points. In the general case, the solutions can be complex. We establish a relation between the functional form of the nonlinearity of the Klein–Gordon equations and the functional dependence of the solutions on rivertons and their derivatives. We study the conditions under which the nonlinearity of the Klein–Gordon equation has a specific functional form and present examples. We establish a relation between the geometric structure of rivertons and the initial conditions.

Keywords: multidimensional nonlinear Klein–Gordon equation, multidimensional quasilinear first-order equation, exact solution, riverton.

Funding Agency Grant Number
Russian Foundation for Basic Research 16-42-732119 р_офи_м
Ministry of Education and Science of the Russian Federation 3.2111.2017/4.6
This work was supported by the Ministry of Education and Science of the Russian Federation (State Order and Project No. 3.2111.2017/4.6) and the Russian Foundation for Basic Research (Grant No. 16-42-732119_a_ofi_m) and in part by the state program of Kazan University aimed at its performance among the world's leading research and education centers.


DOI: https://doi.org/10.4213/tmf9502

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English version:
Theoretical and Mathematical Physics, 2018, 197:3, 1701–1713

Bibliographic databases:

PACS: 03.65.Ge, 03.65.Pm, 02.30.Jr
MSC: 35L70, 35L40, 81Q05
Received: 02.11.2017
Revised: 30.05.2018

Citation: V. M. Zhuravlev, “Multidimensional nonlinear Klein–Gordon equations and rivertons”, TMF, 197:3 (2018), 356–370; Theoret. and Math. Phys., 197:3 (2018), 1701–1713

Citation in format AMSBIB
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