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This article is cited in 6 scientific papers (total in 6 papers)
Multidimensional quasilinear first-order equations and multivalued solutions of the elliptic and hyperbolic equations
V. M. Zhuravlev Technological Research Institute of Ulyanovsk State University, Ulyanovsk, Russia
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.
Received: 06.03.2015 Revised: 09.06.2015
Citation:
V. M. Zhuravlev, “Multidimensional quasilinear first-order equations and multivalued solutions of the elliptic and hyperbolic equations”, TMF, 186:3 (2016), 371–385; Theoret. and Math. Phys., 186:3 (2016), 320–332
Linking options:
https://www.mathnet.ru/eng/tmf8889https://doi.org/10.4213/tmf8889 https://www.mathnet.ru/eng/tmf/v186/i3/p371
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Abstract page: | 523 | Full-text PDF : | 155 | References: | 52 | First page: | 37 |
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