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+13+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.
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
This publication is cited in the following 6 articles:
V. M. Zhuravlev, “Tochnye singulyarnye resheniya uravnenii Khokhlova - Zabolotskoi i kvazilineinye uravneniya pervogo poryadka”, Izvestiya vysshikh uchebnykh zavedenii. Povolzhskii region. Fiziko-matematicheskie nauki, 2024, no. 1, 160–174
V. M. Zhuravlev, “Nelineinye modeli volnovykh protsessov v razmernosti (1 + 1) i kvazilineinye uravneniya pervogo poryadka”, Izvestiya vysshikh uchebnykh zavedenii. Povolzhskii region. Fiziko-matematicheskie nauki, 2023, no. 3, 143–158
V. M. Zhuravlev, V. M. Morozov, “Mnogoznachnye resheniya mnogomernykh lineinykh uravnenii teploprovodnosti i rivertony”, Izvestiya vysshikh uchebnykh zavedenii. Povolzhskii region. Fiziko-matematicheskie nauki, 2021, no. 2, 90–104
V M Zhuravlev, “The principle of materiality of space and the theory of fundamental fields”, J. Phys.: Conf. Ser., 2081:1 (2021), 012038
V. M. Zhuravlev, “Multidimensional nonlinear Klein–Gordon equations and rivertons”, Theoret. and Math. Phys., 197:3 (2018), 1701–1713
V. M. Zhuravlev, V. M. Morozov, “Mnogoznachnye resheniya uravnenii diffuzii i gidrodinamika”, Izvestiya vysshikh uchebnykh zavedenii. Povolzhskii region. Fiziko-matematicheskie nauki, 2018, no. 3, 87–110