Smooth solutions for a ${p}$-system of mixed type

TL;DR

This paper investigates smooth solutions of a mixed-type p-system, showing that periodic smooth solutions must be constant and that solutions with hyperbolic initial data develop shocks, highlighting the system's complex behavior.

Contribution

It provides the first analysis of smooth solutions for a mixed-type p-system without assuming hyperbolicity, revealing conditions under which solutions are constant or develop shocks.

Findings

Periodic smooth solutions are necessarily constant.

Solutions with hyperbolic initial data cannot be extended indefinitely and develop shocks.

The system connects to integrable systems through a reduction of the Benney moments chain.

Abstract

In this note we analyze smooth solutions of a $p$-system of the \textit{mixed} type. Motivating example for this is a 2-components reduction of the Benney moments chain which appears to be connected to theory of integrable systems. We don't assume a-priory that the solutions in question are in the Hyperbolic region. Our main result states that the only smooth solutions of the system which are periodic in $x$ are necessarily constants. As for initial value problem we prove that if the initial data is strictly hyperbolic and periodic in $x$ then the solution can not extend to $[t_0;+\infty)$ and shocks are necessarily created.