Jordan form, parabolicity and other features of change of type transition for hydrodynamic type systems

TL;DR

This paper investigates the transition of two-component hydrodynamic systems between hyperbolic and elliptic types, focusing on Jordan form structures, parabolic hodograph equations, and conditions affecting these transitions, with applications to various Hamiltonian systems.

Contribution

It characterizes the Jordan form and parabolicity of transition lines in hydrodynamic systems and analyzes conditions for type change, including numerical results for specific equations.

Findings

Hydrodynamic systems assume Jordan form with 2x2 Jordan block on transition lines.

Hodograph equations become parabolic during type transitions.

Numerical results illustrate crossing of transition line in dispersionless Boussinesq equation.

Abstract

Changes of type transitions for the two-component hydrodynamic type systems are discussed. It is shown that these systems generically assume the Jordan form (with 2 X 2 Jordan block) on the transition line with hodograph equations becoming parabolic. Conditions which allow or forbid the transition from hyperbolic domain to elliptic one are discussed. Hamiltonian systems and their special subclasses and equations, like dispersionless nonlinear Schroedinger, dispersionless Boussinesq, one-dimensional isentropic gas dynamics equations and nonlinear wave equations are studied. Numerical results concerning the crossing of transition line for the dispersionless Boussinesq equation are presented too.