Dispersionful Version of WDVV Associativity System

TL;DR

This paper introduces a scaling method for the WDVV associativity system that creates a hierarchy with both dispersionless and dispersive integrable structures, expanding understanding of Hamiltonian formulations.

Contribution

It proposes a novel scaling procedure that transforms third order Hamiltonian structures into non-homogeneous forms, enabling new integrable hierarchies with dispersion effects.

Findings

Constructed a hierarchy with both dispersionless and dispersive limits.

Demonstrated at least two dispersive integrable extensions of the system.

Connected the scaling procedure to existing Hamiltonian structures in WDVV equations.

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)