On Laplace Invariants of Two-Dimensional Hyperbolic Equation with Mixed Derivative and Quadratic Nonlinearities

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Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system's guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	DOI: 10.46698/m1855-1369-1428-v On Laplace Invariants of Two-Dimensional Hyperbolic Equation with Mixed Derivative and Quadratic Nonlinearities Rakhmelevich, I. V. Vladikavkaz Mathematical Journal 2024. Vol. 26. Issue 2. Abstract: We study two-dimensional nonlinear hyperbolic equation of the second order with variable coefficients. The left side of this equation contains quadratic nonlinearities on unknown function and its derivatives. We consider a set of linear multiplicative transformations of unknown function which keep a form of initial equation. By analogy with linear equations, the Laplace invariants are determined as the invariants of this transformation. Expressions for the Laplace invariants over the coefficients of the equation and their first derivatives are obtained. We consider both the general case and the case when some coefficients of the equation equals to zero. The main theorem about Laplace invariants is proved. According to this theorem, two nonlinear hyperbolic equations of the considering form can be connected with the help of linear multiplicative transformation if only if the Laplace invariants for both equations have the same values. We have found the equivalent systems of the first order equations, containing the Laplace invariants, for considering equation in general case and in the case when some coefficients of the equation equals to zero. It is shown that the solution of the initial equation can be received in quadratures if some additional conditions on the coefficients and on the Laplace invariants are fulfilled. Keywords: partial differential equation, hyperbolic equation, Laplace invariant, linear multiplicative transformation, quadratic nonlinearity Language: Russian Download the full text For citation: Rakhmelevich, I. V. On Laplace Invariants of Two-Dimensional Hyperbolic Equation with Mixed Derivative and Quadratic Nonlinearities, Vladikavkaz Math. J., 2024, vol. 26, no. 2, pp. 113-121 (in Russian). DOI 10.46698/m1855-1369-1428-v + References 1. Goursat, E. Cours d'Analyse Mathematique, Paris, 1933. 2. Tricomi, F. Lectures on Partial Differential Equations, Moscow, IL, 1957 (in Russian). 3. Dzhokhadze, O. M. Laplace Invariants for Some Classes of Linear Partial Differential Equations, Differential Equations, 2004, vol. 40, no. 1, pp. 63-74. DOI: 10.1023/B:DIEQ.0000028714.62481.2d. 4. Mironov, A. N. and Mironova, L. B. On Laplace Invariants for Equations with Dominating Third-Order Partial Derivative and Two Independent Variables, Mathematical Notes, 2016, vol. 99, no. 1-2, pp. 110-115. DOI: 10.1134/S0001434616010119. 5. Mironov, A. N. and Mironova, L. B. Laplace Invariants for a Fourth-Order Equation with Two Independent Variables, Russian Mathematics, 2014, vol. 58, no. 10, pp. 22-28. DOI: 10.3103/S1066369X14100041. 6. Mironov, A. N. and Mironova, L. B. Laplace Invariants of an Equation with a Dominating Partial Derivative and Three Independent Variables, Differential Equations, 2019, vol. 55, no. 1, pp. 68-74. DOI: 10.1134/S0012266119010075. 7. Kuznetsova, M. N. Laplace Transformation and Nonlinear Hyperbolic Equations, Ufimskii Matematicheskii Zhurnal, 2009, vol. 1, no. 3, pp. 87-96 (in Russian). 8. Zhiber, А. V. and Sokolov, V. V. Exactly Integrable Hyperbolic Equations of Liouville Type, Russian Mathematical Surveys, 2001, vol. 56, no. 1, pp. 61-101. DOI: 10.1070/RM2001v056n01ABEH000357. 9. Startsev, S. Y. Laplace Invariants of Hyperbolic Equations Linearizable by a Differential Substitution, Theoretical and Mathematical Physics, 1999, vol. 120, no. 2, pp. 1009-1018. DOI: 10.1007/BF02557408. The sole copyright holder of the published work is the Founder of the Vladikavkaz Mathematical Journal. The terms of use of this work are governed by an open license (Creative Commons Attribution-NonCommercial 4.0 International). The use of metadata of the scientific article, including the title, abstract, author information, references, identifiers, and other bibliographic description elements for subsequent unrestricted use, is carried out under the terms of the CC BY or CC0 open licenses. ← Contents of issue


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