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EXTENDED SYMMETRY OF THE WITTEN--DIJKGRAAF--VERLINDE--VERLINDE EQUATION OF MONGE--AMPERE TYPE.

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Title: EXTENDED SYMMETRY OF THE WITTEN--DIJKGRAAF--VERLINDE--VERLINDE EQUATION OF MONGE--AMPERE TYPE.
Authors: Sitko, Patryk1 psitko@agh.edu.pl, Tsyfra, Ivan1 tsyfra@agh.edu.pl
Source: Opuscula Mathematica. 2025, Vol. 45 Issue 2, p251-274. 24p.
Subject Terms: *TRANSFORMATION groups, *NONLINEAR differential equations, *ORDINARY differential equations, *PARTIAL differential equations, *DIFFERENTIAL equations
Abstract: 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. [ABSTRACT FROM AUTHOR]
Database: Academic Search Index
Description
Abstract: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. [ABSTRACT FROM AUTHOR]
ISSN:12329274
DOI:10.7494/OpMath.2025.45.2.251