Multivariate Toda hierarchies and biorthogonal polynomials

Abstract

A new multivariate Toda hierarchy of nonlinear partial differential equations adapted to multivariate biorthogonal polynomials is discussed. This integrable hierarchy is associated with non-standard multivariate biorthogonality. Wave and Baker functions, linear equations, Lax and Zakharov–Shabat equations, KP type equations, appropriate reductions, Darboux or linear spectral transformations, and bilinear equations involving linear spectral transformations are presented.