Opuscula Math. 45, no. 2 (2025), 251-274
https://doi.org/10.7494/OpMath.2025.45.2.251
Opuscula Mathematica
Extended symmetry of the Witten-Dijkgraaf-Verlinde-Verlinde equation of Monge-Ampere type
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.
Keywords: non-point symmetries, Witten-Dijkgraaf-Verlinde-Verlinde equation, symmetry group, transformations, Lie algebra.
Mathematics Subject Classification: 35B06, 35A22.
- G.W. Bluman, J.D. Cole, The general similarity solution of the heat equation, J. Math. Mech. 18 (1969), 1025-1042. https://doi.org/10.1512/iumj.1969.18.18074
- G.W. Bluman, S. Kumei, Symmetries and Differential Equations, Springer-Verlag, New York, 1981.
- G.W. Bluman, Z. Yang, A symmetry-based method for constructing nonlocally related partial differential equation systems, J. Math. Phys. 54 (2013), 093504. https://doi.org/10.1063/1.4819724
- R. Conte, M.L. Gandarias, Symmetry reductions of a particular set of equations of associativity in two-dimensional topological field theory, J. Phys. A: Math. Gen. 38 (2005), 1187-1196. https://doi.org/10.1088/0305-4470/38/5/018
- R. Dijkgraaf, H. Verlinde, E. Verlinde, Topological strings in \(d<1\), Nucl. Phys. B 352 (1991), 59-86. https://doi.org/10.1016/0550-3213(91)90129-l
- R.K. Dodd, R.K. Bullogh, Bäcklund transformations for sine-Gordon equations, Proc. R. Soc. Lond. A 351 (1976), 499-523. https://doi.org/10.1098/rspa.1976.0154
- B. Dubrovin, Geometry of 2D topological field theories, [in:] R. Donagi, B. Dubrovin, E. Frenkel, E. Previato, Integrable Systems and Quantum Groups, Lecture Notes in Mathematics, vol. 1620, Montecatini Terme, Italy, 1993, 120-348. https://doi.org/10.1007/bfb0094793
- B.A. Dubrovin, S.P. Novikov, Hydrodynamics of weakly deformed soliton lattices. Differential geometry and Hamilton theory, Russ. Math. Surv. 44 (1989), 35-124. https://doi.org/10.1070/rm1989v044n06abeh002300
- E.V. Ferapontov, On integrability of \(3 \times 3\) semi-Hamiltonian hydrodynamic type systems which do not possess Riemann invariants, Physica D: Nonlinear Phenomena 63 (1993), 50-70. https://doi.org/10.1016/0167-2789(93)90146-r
- E.V. Ferapontov, On the matrix Hopf equation and integrable Hamiltonian systems of hydrodynamic type, which do not possess Riemann invariants, Physics Letters A 179 (1993), 391-397. https://doi.org/10.1016/0375-9601(93)90096-i
- E.V. Ferapontov, Several conjectures and results in the theory of integrable Hamiltonian systems of hydrodynamic type, which do not possess Riemann invariants, Theor. Math. Phys. 99 2 (1994), 567-570. https://doi.org/10.1007/bf01016140
- E.V. Ferapontov, Dupin hypersurfaces and integrable hamiltonian systems of hydrodynamic type, which do not possess Riemann invariants, Diff. Geom. Appl. 5 (1995), 121-152. https://doi.org/10.1016/0926-2245(95)00011-r
- E.V. Ferapontov, Hypersurfaces with flat centroaffine metric and equations of associativity, Geometriae Dedicata 103 (2004), 33-49. https://doi.org/10.1023/b:geom.0000013839.59173.a6
- E.V. Ferapontov, O.I. Mokhov, Equations of associativity in two-dimensional field theory as integrable Hamiltonian nondiagonalizable systems of hydrodynamic type, Functional Analysis and Applications 30 (1996), 195-203. https://doi.org/10.1007/bf02509506
- E.V. Ferapontov, C.A.P. Galvão, O.I. Mokhov, Y. Nutku, Bi-Hamiltonian Structure in 2-d Field Theory, Commun. Math. Phys. 186 (1997), 649-669. https://doi.org/10.1007/s002200050123
- E.V. Ferapontov, L. Hadjikos, K.R. Khusnutdinova, Integrable equations of the dispersionless Hirota type and hypersurfaces in the Lagrangian Grassmannian, International Mathematics Research Notices 3 (2010), 496-535. https://doi.org/10.1093/imrn/rnp134
- W.I. Fushchych, I.M. Tsifra, On a reduction and solutions of nonlinear wave equations with broken symmetry, J. Phys. A 20 (1987), L45-L48. https://doi.org/10.1088/0305-4470/20/2/001
- A.V. Kiselev, Methods of geometry of differential equations in analysis of integrable models of field theory, Fundamentalnaya i Prikladnaya Matematika 10 (2004), 57-165.
- O.I. Mokhov, Symplectic and Poisson geometry on loop spaces of manifolds and nonlinear equations, [in:] S.P. Novikov (ed.), Topics in Topology and Mathematical Physics, Transl. Am. Math. Soc., Ser. 2, vol. 170, Am. Math. Soc., Providence, 1995, 121-151. https://doi.org/10.1090/trans2/170/06
- P.J. Olver, Applications of Lie Groups to Differential Equations, 2nd ed., Springer-Verlag, New York, 1993.
- P.J. Olver, P. Rosenau, The construction of special solutions to partial differential equations, Phys. Lett. A 114 (1986), 107-112. https://doi.org/10.1016/0375-9601(86)90534-7
- L.V. Ovsiannikov, Group Analysis of Differential Equations, Academic Press, New York, 1982.
- M.V. Pavlov, R.F. Vitolo, On the bi-Hamiltonian geometry of WDVV equations, Lett. Math. Phys. 105 (2015), 1135-1163. https://doi.org/10.1007/s11005-015-0776-8
- S.P. Tsarëv, The geometry of Hamiltonian systems of hydrodynamic type. The generalized hodograph method, Math. USSR Izv. 37 (1991), 397-419. https://doi.org/10.1070/im1991v037n02abeh002069
- I. Tsyfra, Non-local ansätze for nonlinear heat and wave equations, J. Phys. A: Math. Gen. 30 (1997), 2251-2262. https://doi.org/10.1088/0305-4470/30/6/042
- I. Tsyfra, A. Napoli, A. Messina, V. Tretynyk, On new ways of group methods for reduction of evolution-type equations, J. Math. Anal. Appl. 307 (2005), 724-735. https://doi.org/10.1016/j.jmaa.2005.03.075
- J. Vašíček, R. Vitolo, WDVV equations and invariant bi-Hamiltonian formalism, J. High Energ. Phys. 129 (2021). https://doi.org/10.1007/jhep08(2021)129
- E. Witten, On the structure of the topological phase of two-dimensional gravity, Nucl. Phys. B 340 (1990), 281-332. https://doi.org/10.1016/0550-3213(90)90449-n
- R.Z. Zhdanov, I.M. Tsyfra, R.O. Popovych, A precise definition of reduction of partial differential equations, J. Math. Anal. Appl. 238 (1999), 101-123. https://doi.org/10.1006/jmaa.1999.6511
- Patryk Sitko
https://orcid.org/0000-0002-2510-6528- AGH University of Krakow, Faculty of Applied Mathematics, al. Mickiewicza 30, 30-059 Krakow, Poland
- Ivan Tsyfra (corresponding author)
- https://orcid.org/0000-0001-6665-3934
- AGH University of Krakow, Faculty of Applied Mathematics, al. Mickiewicza 30, 30-059 Krakow, Poland
- Communicated by Aleksander Gomilko.
- Received: 2024-04-03.
- Revised: 2025-01-19.
- Accepted: 2025-01-28.
- Published online: 2025-03-10.
