Abstract:
We discuss an extension of the theory of multidimensional second-order equations of the elliptic and hyperbolic types related to multidimensional quasilinear autonomous first-order partial differential equations. Calculating the general integrals of these equations allows constructing exact solutions in the form of implicit functions. We establish a connection with hydrodynamic equations. We calculate the number of free functional parameters of the constructed solutions. We especially construct and analyze implicit solutions of the Laplace and d'Alembert equations in a coordinate space of arbitrary finite dimension. In particular, we construct generalized Penrose–Rindler solutions of the d'Alembert equation in $3{+}1$ dimensions.
Keywords:exact solution of multidimensional nonlinear hyperbolic equations, exact solution of multidimensional nonlinear elliptic equations, multivalued solution, system of nonlinear equations of hydrodynamic type, electromagnetic wave equation, Laplace equation, d'Alembert equation.
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