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,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\"odinger hierarchies, the "inverse" hierarchy and equations associated with the real-analytic Eisenstein series E(β,β;1/2)among them. Specific bi-Hamiltonian structure of these equations is also discussed.
0