On the Hamiltonian Representation of the Associativity Equations

1997 ◽  
pp. 75-91 ◽  
Author(s):  
E. V. Ferapontov ◽  
O. I. Mokhov
Keyword(s):  
Associativity Equations

scholarly journals Associativity equations in dispersionless integrable hierarchies

2001 ◽  
Vol 515 (3-4) ◽  
pp. 483-492 ◽  
Author(s):  
A. Boyarsky ◽  
A. Marshakov ◽  
O. Ruchayskiy ◽  
P. Wiegmann ◽  
A. Zabrodin
Keyword(s):  
Integrable Hierarchies ◽  
Associativity Equations

scholarly journals LIOUVILLE INTEGRABLE REDUCTIONS OF THE ASSOCIATIVITY EQUATIONS ON THE SET OF STATIONARY POINTS OF AN INTEGRAL IN THE CASE OF THREE PRIMARY FIELDS

2019 ◽  
Vol 47 (1) ◽  
pp. 88-90
Author(s):  
O.I. Mokhov ◽  
N.A. Strizhova
Keyword(s):  
Stationary Points ◽  
Evolutionary System ◽  
Liouville Integrability ◽  
Russian Science ◽  
Partial Derivatives ◽  
First Order ◽  
System Of Hydrodynamic Type ◽  
Associativity Equations ◽  
Liouville Integrable ◽  
Hamiltonian Property

In this work, in the case of three primary fields, a reduction of the associativity equations (the Witten–Dijkgraaf–Verlinde–Verlinde system, see (Witten, 1990, Dijkgraaf et al., 1991, Dubrovin, 1994) with antidiagonal matrix ηij on the set of stationary points of a nondegenerate integral quadratic with respect to the first-order partial derivatives is constructed in an explicit form and its Liouville integrability is proved. In Mokhov’s paper (Mokhov, 1995, Mokhov, 1998), these associativity equations were presented in the form of an integrable nondiagonalizable system of hydrodynamic type. In the papers (Ferapontov, Mokhov, 1996, Ferapontov et al., 1997, Mokhov, 1998), a bi-Hamiltonian representation for these equations and a nondegenerate integral quadratic with respect to the first-order partial derivatives were found. Using Mokhov’s construction on canonical Hamiltonian property of an arbitrary evolutionary system on the set of stationary points of its nondegenerate integral of the papers (Mokhov, 1984, Mokhov, 1987), we construct explicitly the reduction for the integral quadratic with respect to the first-order partial derivatives, found explicitly the Hamiltonian of the corresponding canonical Hamiltonian system. We also found three functionally-independent integrals in involution with respect to the canonical Poisson bracket on the phase space for the constructed reduction of the associativity equations and thus proved the Liouville integrability of this reduction. This work is supported by the Russian Science Foundation under grant No. 18-11-00316.

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