Extended symmetry of the Witten-Dijkgraaf-Verlinde-Verlinde equation of Monge-Ampere type

• Autorzy: Sitko Patryk , Tsyfra Ivan

Identyfikatory

DOI

10.7494/OpMath.2025.45.2.251

• Języki publikacji: EN

Abstrakty

EN

We construct the Lie algebra of extended symmetry group for the Monge–Ampere type Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) equation. This algebra includes novel generators that are unobtainable within the framework of the classical Lie approach and correspond to non-point group transformation of dependent and independent variables. The expansion of symmetry is achieved by introducing new variables through second-order derivatives of the dependent variable. By integrating the Lie equations, we derive transformations that enable the generation of new solutions to the Witten–Dijkgraaf–Verlinde–Verlinde equation from a known one. These transformations yield formulas for obtaining new solutions in implicit form and Bäcklund-type transformations for the nonlinear associativity equations. We also demonstrate that, in the case under study, introducing a substitution of variables via third-order derivatives, as previously used in the literature, does not yield generators of non-point transformations. Instead, this approach produces only the Lie groups of classical point transformations. Furthermore, we perform a group reduction of partial differential equations in two independent variables to a system of ordinary differential equations. This reduction leads to the explicit solution of the fully nonlinear differential equation. Notably, the symmetry group of non-point transformations expands significantly when this method is applied to the second-order differential equation, resulting in a corresponding infinite-dimensional Lie algebra. Finally, we show that auxiliary variables can be systematically derived within the framework of a generalized approach to symmetry reduction of differential equations.

Słowa kluczowe

EN

non-point symmetries; Witten–Dijkgraaf–Verlinde–Verlinde equation; symmetry group; transformations; Lie algebra

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2025
• Tom: Vol. 45, no. 2
• Strony: 251--274
• Opis fizyczny: Bibliogr. 29 poz., tab.

Twórcy

autor

Sitko Patryk

• AGH University of Krakow, Faculty of Applied Mathematics, al. Mickiewicza 30, 30–059 Krakow, Poland

• https://orcid.org/0000-0002-2510-6528

autor

Tsyfra Ivan

• AGH University of Krakow, Faculty of Applied Mathematics, al. Mickiewicza 30, 30–059 Krakow, Poland

• https://orcid.org/0000-0001-6665-3934

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Uwagi

Opracowanie rekordu ze środków MNiSW, umowa nr POPUL/SP/0154/2024/02 w ramach programu "Społeczna odpowiedzialność nauki II" - moduł: Popularyzacja nauki (2025)