On singular solutions of a multidimensional differential

L. L. Ryskina

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 23 (2019) no. 2, pp. 394-401.

Résumé

In the theory of ordinary differential equations, the Clairaut equation is well known. This equation is a non-linear differential equation unresolved with respect to the derivative. Finding the general solution of the Clairaut equation is described in detail in the literature and is known to be a family of integral lines. However, along with the general solution, for such equations there exists a singular (special) solution representing the envelope of the given family of integral lines. Note that the singular solution of the Clairaut equation is of particular interest in a number of applied problems. In addition to the ordinary Clairaut differential equation, a differential equation of the first order in partial derivatives of the Clairaut type is known. This equation is a multidimensional generalization of the ordinary differential Clairaut equation, in the case when the sought function depends on many variables. The problem of finding a general solution for partial differential equations of the Clairaut is known to be. It is known that the complete integral of the equation is a family of integral (hyper) planes. In addition to the general solution, there may be partial solutions, and, in some cases, it is possible to find a singular solution. Generally speaking, there is no general algorithm for finding a singular solution, since the problem is reduced to solving a system of nonlinear algebraic equations. The article is devoted to the problem of finding a singular solution of Clairaut type differential equation in partial derivatives for the particular choice of a function from the derivatives in the right-hand side. The work is organized as follows. The introduction provides a brief overview of some of the current results relating to the study of Clairaut-type equations in field theory and classical mechanics. The first part provides general information about differential equations of the Clairaut-type in partial derivatives and the structure of its general solution. In the main part of the paper, we discuss the method for finding singular solutions of the Clairaut-type equations. The main result of the work is to find singular solutions of equations containing power and exponential functions.

Détail

Keywords: partial differential equations
Mots-clés : Clairaut-type equations, singular solutions.

Bibliographie

[1] Clairaut A., “Solution de plusieurs problèmes où il s'agit de trouver des Courbes dont la propriété consiste dans une certaine relation entre leurs branches, exprimée par une Équation donnée”, Histoire Acad. R. Sci. Paris (1734), 1736, 196–215 https://gallica.bnf.fr/ark:/12148/bpt6k3531x/f344.table

[2] El'sgol'ts L. E., Differentsial'nye uravneniia i variatsionnoe ischislenie [Differential equations and calculus of variations], Nauka, Moscow, 1969, 424 pp. (In Russian)

[3] Kamke E., Differentialgleichungen. Lösungsmethoden und Lösungen, v. I, Gewöhnliche Differentialgleichungen, B.G. Teubner, Stuttgart, 1977, xxvi+668 pp. (In German) | MR | Zbl

[4] Courant R., Hilbert D., Methods of mathematical physics, v. 2, Partial differential equations, John Wiley Sons, New York, London, 1962, xxii+830 pp. | MR | Zbl

[5] Lavrov P. M., Merzlikin B. S., “Loop expansion of the average effective action in the functional renormalization group approach”, Phys. Rev. D, 92:8 (2015), 085038, arXiv: [hep-th] 1506.04491 | DOI | MR

[6] Lavrov P. M., Merzlikin B. S., “Legendre transformations and Clairaut-type equations”, Phys. Lett. B, 756 (2016), 188–193, arXiv: [hep-th] 1602.04911 | DOI | Zbl

[7] Walker M., Duplij S., “Cho-Duan-Ge decomposition of QCD in the constraintless Clairaut-type formalism”, Phys. Rev. D, 91:6 (2015), 064022, arXiv: [hep-th] 1411.6968 | DOI

[8] Duplij S., “A new Hamiltonian formalism for singular Lagrangian theories”, Journal of Kharkov National University, Ser. Nuclei, Particles and Fields, 969:3 (2011), 34–39, arXiv: 0909.2695 [math-ph]

[9] Zyryanova O. V., Mudruk V. I., “Singular Solutions of Clairaut Equations”, Russ. Phys. J., 61:4 (2018), 635–642 | DOI | Zbl

[10] Rakhmelevich I. V., “On the Solutions of Multi-dimensional Clairaut Equation with Multi-homogeneous Function of the Derivatives”, Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 14:4(1) (2014), 374–381 (In Russian) | Zbl