Elliptic Euler-Poisson-Darboux equation, critical points and integrable systems

TL;DR

This paper investigates the structure of critical points for functions satisfying the elliptic Euler-Poisson-Darboux equation, revealing their connection to integrable hierarchies and bi-Hamiltonian structures.

Contribution

It introduces explicit integrable quasi-linear systems related to the elliptic Euler-Poisson-Darboux equation and explores their variational and differential properties.

Findings

Explicit integrable hierarchies including dispersionless Toda and nonlinear Schrödinger systems

Identification of bi-Hamiltonian structures in these equations

Connection to Eisenstein series and real-analytic functions

Abstract

Structure and properties of families of critical points for classes of functions $W(z,\bar{z})$ obeying the elliptic Euler-Poisson-Darboux equation $E(1/2,1/2)$ are studied. General variational and differential equations governing the dependence of critical points in variational (deformation) parameters are found. Explicit examples of the corresponding integrable quasi-linear differential systems and hierarchies are presented There are the extended dispersionless Toda/nonlinear Schr\"{o}dinger hierarchies, the "inverse" hierarchy and equations associated with the real-analytic Eisenstein series $E(\beta,\bar{{\beta}};1/2)$among them. Specific bi-Hamiltonian structure of these equations is also discussed.