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

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.