Multidimensional Nonlinear Klein–Gordon Equations and Rivertons

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.