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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
V. M. Zhuravlev, Theor. Math. Phys., 174, 236–246 (2013).
V. M. Zhuravlev, Prostrantsvo, vremya i fundamental’nye vzaimodeistviya, no. 4, 56–67 (2013).
B. L. Rozhdestvenskij and N. N. Yanenko, Systems of Quasilinear Equations and Their Applications to Gas Dynamics [in Russian], Nauka, Moscow (1978); English transl. (Transl. Math. Monogr., vol. 55), Amer. Math. Soc., Providence, R. I. (1983).
A. G. Kulikovskii, E. I. Sveshnikova, and A. P. Chugainova, Mathematical Methods for Studying Discontinuous Solutions of Nonlinear Hyperbolic Systems of Equations (Lekts. Kursy NOTs, vol. 16), Steklov Math. Institute, RAS, Moscow (2010).
V. I. Arnold, Singularities of Caustics and Wave Fronts (Math. Its Appl., vol. 62), Kluwer, Dordrecht (1990).
R. Penrose and W. Rindler, Spinors and Space–Time, vol. 2, Spinor and Twistor Methods in Space–Time Geometry, Cambridge Univ. Press, Cambridge (1986); V. V. Kassandrov and J. A. Riscalla, “Particles as singularities within the unified algebraic field dynamics,” in: Geometrization of Physics III (V. I. Bashkov, ed.), Kazan State Univ., Kazan (1997), pp. 199–212; ar**v:gr-qc/9809056v1 (1998).
S. P. Tsarev, Math. USSR-Izv., 37, 397–419 (1991).
E. V. Ferapontov and K. R. Khusnutdinova, Commun. Math. Phys., 248, 187–206 (2004); J. Phys. A, 37, 2949–2963 (2004); ar**v:nlin/0310021v1 (2003); J. Math. Phys., 45, 2365–2377 (2004); ar**v:nlin/0312015v1 (2003).
E. V. Ferapontov, K. R. Khusnutdinova, and M. V. Pavlov, Theor. Math. Phys., 144, 907–915 (2005).
V. M. Zhuravlev, Gravit. Cosmol., 17, 201–217 (2011).
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was supported by the Ministry of Education and Science of the Russia Federation (State Mission and Project No. 14.Z50.31.0015).
__________
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 186, No. 3, pp. 371–385, March, 2016.
Rights and permissions
About this article
Cite this article
Zhuravlev, V.M. Multidimensional quasilinear first-order equations and multivalued solutions of the elliptic and hyperbolic equations. Theor Math Phys 186, 320–332 (2016). https://doi.org/10.1134/S0040577916030028
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0040577916030028