Quasilinear systems of Jordan block type and the mKP hierarchy

Hydrodynamic type systems in Riemann invariants arise in a whole range of applications in fluid dynamics, Whitham averaging procedure, differential geometry and the theory of Frobenius manifolds. In this paper we discuss parabolic (Jordan block) analogues of diagonalisable systems. Our main observation is that integrable quasilinear systems of Jordan block type are parametrised by solutions of the modified Kadomtsev–Petviashvili hierarchy. Such systems appear naturally as degenerations of quasilinear systems associated with multi-dimensional hypergeometric functions, in the context of parabolic regularisation of the Riemann equation, as finite-component reductions of hydrodynamic chains, and as hydrodynamic reductions of linearly degenerate dispersionless integrable PDEs in multi-dimensions.