Geodesic
Multidimensional quasilinear first-order equations and multivalued solutions of the elliptic and hyperbolic equations
Teoretičeskaâ i matematičeskaâ fizika, Tome 186 (2016) no. 3, pp. 371-385 Cet article a éte moissonné depuis la source Math-Net.Ru

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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
Mots-clés : Laplace equation, d'Alembert equation. 
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     title = {Multidimensional quasilinear first-order equations and multivalued solutions of the~elliptic and hyperbolic equations},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
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V. M. Zhuravlev. Multidimensional quasilinear first-order equations and multivalued solutions of the elliptic and hyperbolic equations. Teoretičeskaâ i matematičeskaâ fizika, Tome 186 (2016) no. 3, pp. 371-385. http://geodesic.mathdoc.fr/item/TMF_2016_186_3_a1/

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