The algebraic and Hamiltonian structure of the dispersionless Benney and Toda hierarchies

TL;DR

This paper explores the algebraic and Hamiltonian structures of the dispersionless Benney and Toda hierarchies, revealing their equivalence, explicit formulas for conserved quantities, and extensions to related systems using hypergeometric functions.

Contribution

It introduces a symmetric variable framework that simplifies calculations and establishes the equivalence between the Benney and Toda hierarchies, extending results to rational and logarithmic Lax functions.

Findings

Explicit formulas for conserved charges and fluxes

Demonstration of the equivalence between Benney and Toda hierarchies

Extension to systems with rational and logarithmic Lax functions

Abstract

The algebraic and Hamiltonian structures of the multicomponent dispersionless Benney and Toda hierarchies are studied. This is achieved by using a modified set of variables for which there is a symmetry between the basic fields. This symmetry enables formulae normally given implicitly in terms of residues, such as conserved charges and fluxes, to be calculated explicitly. As a corollary of these results the equivalence of the Benney and Toda hierarchies is established. It is further shown that such quantities may be expressed in terms of generalized hypergeometric functions, the simplest example involving Legendre polynomials. These results are then extended to systems derived from a rational Lax function and a logarithmic function. Various reductions are also studied.