Dispersionful Version of WDVV Associativity System

Abstract

B.A. Dubrovin proved that remarkable WDVV associativity equations are integrable systems. In a simplest nontrivial three-component case these equations can be written as a nondiagonalizable hydrodynamic type system equivalent to a symmetric reduction of the three wave interaction and to the matrix Hopf equation. Then E.V. Ferapontov and O.I. Mokhov found a local Hamiltonian structure. Finally E.V. Ferapontov, C.A.P. Galv\~{a}o, O.I. Mokhov, Ya. Nutku found a second local Hamiltonian structure. Both local Hamiltonian structure are homogeneous of first and third order (respectively) of Dubrovin--Novikov type. In our paper we suggest a special scaling procedure for independent variables applicable for homogeneous nonlinear PDE's, which allows to incorporate an auxiliary parameter $\epsilon $, such that a corresponding \textquotedblleft intermediate\textquotedblright\ system possesses two remarkable limits: a high-frequency limit ($\epsilon \rightarrow \infty $) back to the original system and a dispersionless limit ($\epsilon \rightarrow 0$) which yields diagonalizable integrable hydrodynamic type system. This means that our procedure allows to transform a homogeneous third order local Hamiltonian structure to non-homogeneous of third order. Thus we create an integrable hierarchy equipped by a pair of local Hamiltonian structures, which (both of them) possess a dispersionless limit. Also we show that this bi-Hamiltonian diagonalizable hydrodynamic type system possesses at least two different dispersive integrable extensions (in a framework of B.A. Dubrovin's approach)