Multidimensional Nonlinear Klein–Gordon Equations and Rivertons
- 5 Downloads
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.
Keywordsmultidimensional nonlinear Klein–Gordon equation multidimensional quasilinear first-order equation exact solution riverton
Unable to display preview. Download preview PDF.
- 6.Yu. P. Rybakov and V. I. Sanyuk, Multidimensional Solitons [in Russian], RUDN, Moscow (2001).Google Scholar
- 17.V. M. Zhuravlev and V. M. Morozov, “On multi-valued solutions of two-dimensional linear parabolic equations [in Russian],” in: Differential Equations and Their Applications in Mathematical Modeling (XIII International Scientific Conference, Saransk, 12–16 July 2017), SVMO, Saransk (2017), pp. 330–340; http://conf.svmo.ru/files/deamm2017/papers/paper46.pdf (2017).Google Scholar