Abstract
We demonstrate that for an arbitrary number n of primary fields the equations of the associativity can be rewritten in the form of (n — 2) pairwise commuting systems of hydrodynamic type, which appear to be nondiagonalizable but integrable. We propose a natural Hamiltonian representation of the systems under study in the cases n = 3 and n = 4.
This work was partially supported by the International Science Foundation (Grant 93-0166 — O.I.M.). and Grant No. 93-011-168 — E.V.F.) and the INTAS (Grant No. 93-0166 — O.I.M.).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Dubrovin, B., Geometry of 2D topological field theories, preprint SISSA- 89/94/FM, SISSA, Trieste (1994).
Mokhov, O.I., Differential equations of associativity in 2D topological field theories and geometry of nondiagonalizable systems of hydrodynamic type, In: Abstracts of Internat. Conference on Integrable Systems “Nonlinearity and Integrability: from Mathematics to Physics”, February 21-24, 1995, Montpellier, France (1995).
Mokhov, O.I., Symplectic and Poisson geometry on loop spaces of manifolds and nonlinear equations, AMS Translations, series 2 Vol. 170 Topics in topology and mathematical physics, S.P. Novikov, ed., (1995); hep-th/9503076.
Tsarev, S.P., Geometry of Hamiltonian systems of hydrodynamic type, The generalized hodograph method, Izvestiya Akad. Nauk SSSR, Ser. mat. 54 (5) (1990), 1048 - 1068.
Dubrovin, B.A. and Novikov, S.P., Hydrodynamics of weakly deformed soliton lattices. Differential geometry and Hamiltonian theory, Uspekhi Mat Nauk 44 (6) (1989), 29 - 98.
Ferapontov, E.V., On integrability of 3 x 3 semi-Hamiltonian hydrodynamic type systems which do not possess Riemann invariants, Physica D 63 (1993), 50 - 70.
Ferapontov, E.V., On the matrix Hopf equation and integrable Hamiltonian systems of hydrodynamic type, which do not possess Riemann invariants, Phys. Lett. A 179 (1993), 391 - 397.
Ferapontov, E.V., Dupin hypersurfaces and integrable Hamiltonian systems of hydrodynamic type which do not possess Riemann invariants, Diff. Geometry and its Appl. 5 (1995), 121 - 152.
Ferapontov, E.V., Several conjectures and results in the theory of integrable Hamiltonian systems of hydrodynamic type, which do not possess Riemann invariants, Teor. and Mat. Physics 99 (2) (1994), 257 - 262.
Witten, E., On the structure of the topological phase of two-dimensional gravity, Nucl. Physics B 340 (1990), 281 - 332.
Dijkgraaf, R., Verlinde, H., and Verlinde, E., Topological string in gravity, Nucl. Physics B 352 (1991), 59-86.
Witten, E., Two-dimensional gravity and intersection theory on moduli space, Surveys in Diff. Geometry 1 (1991), 243 - 310.
Dubrovin, B., Integrable systems in topological field theory, Nucl. Physics B 379 (1992), 627 - 689.
Mokhov, O.I. and Ferapontov, E.V., On the nonlocal Hamiltonian operators of hydrodynamic type, connected with constant curvature metrics, Uspekhi Mat. Nauk 45 (3) (1990), 191 - 192.
Ferapontov, E.V., Differential geometry of nonlocal Hamiltonian operators of hydrodynamic type, Funkts. Analiz i ego Prilozh. 25 (3) (1991), 37 - 49.
Mokhov, O.I., Hamiltonian systems of hydrodynamic type and constant curvature metrics, Phys. Letters A 166 (3-4) (1992), 215 - 216.
Mokhov, O.I. and Nutku, Y., Bianchi transformation between the real hyperbolic Monge-Ampere equation and the Born-Infeld equation, Letters in Math. Phys. 32 (2) (1994), 121 - 123.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Additional information
Dedicated to the memory of Irina Dorfman
Rights and permissions
Copyright information
© 1997 Birkhäuser Boston
About this chapter
Cite this chapter
Ferapontov, E.V., Mokhov, O.I. (1997). On the Hamiltonian Representation of the Associativity Equations. In: Fokas, A.S., Gelfand, I.M. (eds) Algebraic Aspects of Integrable Systems. Progress in Nonlinear Differential Equations and Their Applications, vol 26. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-2434-1_4
Download citation
DOI: https://doi.org/10.1007/978-1-4612-2434-1_4
Publisher Name: Birkhäuser Boston
Print ISBN: 978-1-4612-7535-0
Online ISBN: 978-1-4612-2434-1
eBook Packages: Springer Book Archive