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

Voir la notice de l'article provenant de la source Math-Net.Ru

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. 
@article{TMF_2016_186_3_a1,
     author = {V. M. Zhuravlev},
     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},
     pages = {371--385},
     publisher = {mathdoc},
     volume = {186},
     number = {3},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2016_186_3_a1/}
}
TY  - JOUR
AU  - V. M. Zhuravlev
TI  - Multidimensional quasilinear first-order equations and multivalued solutions of the~elliptic and hyperbolic equations
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2016
SP  - 371
EP  - 385
VL  - 186
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TMF_2016_186_3_a1/
LA  - ru
ID  - TMF_2016_186_3_a1
ER  - 
%0 Journal Article
%A V. M. Zhuravlev
%T Multidimensional quasilinear first-order equations and multivalued solutions of the~elliptic and hyperbolic equations
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2016
%P 371-385
%V 186
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TMF_2016_186_3_a1/
%G ru
%F TMF_2016_186_3_a1
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/