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

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 Schrödinger 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.

18 pages, no figures

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