On Exact Multidimensional Solutions of a
Nonlinear System of First Order Partial Differential
This study is concerned with a system of two nonlinear first order partial differential equations. The right-hand sides of the system contain the squares of the gradients of the unknown functions. Such type of Hamilton-Jacobi like equations are considered in mechanics and control theory. In the paper, we propose to search a solution in the form of an ansatz, the latter containing a quadratic dependence on the spatial variables and arbitrary functions of time. The use of this ansatz allows us to decompose the search of the solution’s components depending on the spatial variables and time. In order to find the dependence on the spatial variables one needs to solve an algebraic system of some matrix and vector equations and of a scalar equation. A general solution of this system of equations is found in a parametric form. To find the time-dependent components of the solution of the original system, we are faced with a system of nonlinear differential equations. The existence of exact solutions of a certain kind for the original system is established. A number of examples of the constructed exact solutions, including periodic in time and anisotropic in the spatial variables ones, are given. The spatial structure of the solutions is analyzed revealing that it depends on the rank of the matrix of the quadratic form entering the solution.
Alexander Kosov, Leading Researcher, Matrosov Institute for System Dynamics and Control Theory of SB RAS, Post Box 292, 134, Lermontov Str., Irkutsk, 664033, Russian Federation; tel.: (3952) 427100, e-mail: kosov firstname.lastname@example.org
Edward Semenov, Senior Researcher, Matrosov Institute for System Dynamics and Control Theory of SB RAS, Post Box 292, 134, Lermontov Str., Irkutsk, 664033, Russian Federation; tel.: (3952) 453099, e-mail: email@example.com
Vladimir Tirskikh, Associate Professor, Irkutsk State Transport University, 15, Chernyshevsky str., Irkutsk, 664074, Russian Federation; tel.: (3952) 638311, e-mail: tirskikh firstname.lastname@example.org
Kosov A.A., Semenov E.I., Tirskikh V.V. On Exact Multidimensional Solutions of a Nonlinear System of First Order Partial Differential Equation. The Bulletin of Irkutsk State University. Series Mathematics, 2019, vol. 28, pp. 53-68. https://doi.org/10.26516/1997-7670.2019.28.53
- Fushhich V.I., Shtelen V.M., Serov N.I. Simmetrijnyj analiz i tochnye reshenij nelinejnyx uravnenij matematicheskoj fiziki [Symmetry analysis and exact solutions of nonlinear equations of mathematical physics]. Kiev, Naukova dumka, 1989, 336 p. (in Russian)
- Galaktionov V.A. , Posashkov S.A. New exact solutions of parabolic equations with quadratic nonlinearities. Comput. Math. Math. Phys., 1989, vol. 29, no. 2, pp. 112–119.
- Galactionov V.A., Svirshchevskii S.R. Subspaces of nonlinear partial differential equations in mechanics and physics. Chapman & Hall/CRC, 2007, 493 p.
- Gantmaxer F.R. Teoriya matric [Matrix Theory]. Moscow, Nauka Publ., 1988, 552 p. (in Russian)
- Kosov A.A., Semenov E.I. Multidimensional exact solutions to the reaction-diffusion system with power-law nonlinear terms. Siberian Math. J., 2017, vol. 58, no. 4, pp. 619-632. https://doi.org/10.1134/S0037446617040085
- Kosov A.A., Semenov E.I. On Exact Multidimensional Solutions of a Nonlinear System of Reaction–Diffusion Equations / Differential Equations, 2018, vol.54, no 1, pp. 106-120. https://doi.org/10.1134/S0012266118010093
- Kurzhanski A.B. Comparison principle for equations of the Hamilton-Jacobi type in control theory. Proceedings of the Steklov Institute of Mathematics, 2006, vol. 253, supplement 1, pp. 185-195. https://doi.org/10.1134/S0081543806050130
- Markov Y., Rudykh G., Sidorov N., Sinitsyn A., Tolstonogov D. Steady state solutions of the Vlasov-Maxwell system and their stability. Acta Appl. Math., 1992, vol. 28, no. 3. pp. 253-293.
- Polyanin A.D., Zajcev V.F., Zhurov A.I. Metody resheniya nelinejnyx uravnenij matematicheskoj fiziki i mexaniki [Methods for solving nonlinear equations of mathematical physics and mechanics]. Moscow, FIZMATLIT, 2005, 256 p. (in Russian)
- Rudykh G.A., Semenov E.I. The construction of exact solutions of the multidimensional quasilinear heat-conduction equation. Comput. Math. Math. Phys., 1993, vol. 33, no. 8, pp. 1087–1097.
- Rudykh G.A., Semenov E.I. Exact nonnegative solutions of the multidimensional nonlinear diffusion equation. Siberian Math. J., 1998, vol. 39, no. 5, pp. 977-985. https://doi.org/10.1007/BF02672920
- Rudykh G.A., Semenov E.I. Non-self-similar solutions of multidimensional nonlinear diffusion equations. Math. Notes, 2000, vol. 67, no. 2, pp. 200–206. https://doi.org/10.1007/BF02686247
- Sidorov N.A., Sinitsyn A.V. Stacionarnaya sistema Vlasova – Maksvella v ogranichennyx oblastyax. [The stationary Vlasov – Maxwell system in bounded domains]. Nelinejnyj analiz i nelinejnye differencialnye uravneniya [Nonlinear analysis and nonlinear differential equations]. Moscow, Fizmatlit Publ., 2003, pp. 50-88. (in Russian)
- Skubachevskii A.L. Vlasov – Poisson equations for a two-component plasma in homogeneous magnetic field. Russian Math. Surveys, 2014, vol. 69, no. 2, pp. 291- 335. http://dx.doi.org/10.1070/RM2014v069n02ABEH004889
- Titov S.S. Metod konechnomernyh kolec dlya resheniya nelinejnyh uravnenij matematicheskoj fiziki [The method of finite-dimensional rings for solving nonlinear equations of mathematical physics] Aerodinamika, Saratov, Saratovskij Univ. Publ., 1988, pp. 104-110. (in Russian)
- Zajcev V.F., Polyanin A.D. Spravochnik po differencialnym uravneniyam s chastnymi proizvodnymi pervogo poryadka [Handbook of differential equations with partial derivatives of the first order]. Moscow, Fizmatlit Publ., 2003, 416 p. (in Russian)