Critical points, Lauricella functions and Whitham-type equations

TL;DR

This paper explores the connection between semi-Hamiltonian hydrodynamic systems, Lauricella functions, and Whitham equations, revealing how classical integrable equations can be derived through Darboux transformations involving Lauricella functions.

Contribution

It demonstrates that many hydrodynamic type systems are related to critical points of functions satisfying Darboux equations, linking Lauricella functions to integrable Whitham equations.

Findings

Derivation of Whitham equations via Lauricella functions

Identification of semi-Hamiltonian systems as critical point equations

Highlighting the role of Euler-Poisson-Darboux equations

Abstract

A large class of semi-Hamiltonian systems of hydrodynamic type is interpreted as the equations governing families of critical points of functions obeying the classical linear Darboux equations for conjugate nets.The distinguished role of the Euler-Poisson-Darboux equations and associated Lauricella-type functions is emphasised. In particular, it is shown that the classical g-phase Whitham equations for the KdV and NLS equations are obtained via a g-fold iterated Darboux-type transformation generated by appropriate Lauricella functions.