Quasilinear differential constraints for parabolic systems of Jordan-block type

TL;DR

This paper establishes that linear degeneracy is essential for Jordan-block parabolic systems to admit compatible quasilinear constraints, linking it to Hamiltonian properties and providing explicit solutions for specific hierarchies.

Contribution

It proves the necessity and sufficiency of linear degeneracy for 2x2 systems to have quasilinear constraints and connects this to Hamiltonian structures.

Findings

Linear degeneracy is necessary for compatible quasilinear constraints.

For 2x2 systems, linear degeneracy is also sufficient.

Explicit solutions for hierarchies from associativity and delta-functional reduction are provided.

Abstract

We prove that linear degeneracy is a necessary conditions for systems in Jordan-block form to admit a compatible quasilinear differential constraint. Such condition is also sufficient for 2x2 systems and turns out to be equivalent to possess the Hamiltonian property. Some explicit solutions of parabolic systems are herein given: two principal hierarchies arising from the associativity theory and the delta-functional reduction of the El's equation in the hard rod case are integrated.