Wave Interaction For A System In Elastodynamics With Initial Data Lying On The Level Set Of One Of The Riemann Invariants

TL;DR

This paper derives explicit solutions for wave interactions in a hyperbolic elastodynamics system with initial data on Riemann invariant level sets, without smallness restrictions, advancing the understanding of wave behavior in elastic media.

Contribution

It provides explicit formulas for wave interactions in a nonconservative hyperbolic system without assuming small initial data.

Findings

Explicit solution formulas for wave interactions.

Analysis applicable to large initial data.

Advances in understanding wave behavior in elastic systems.

Abstract

This paper is concerned with the study of interaction of waves originating from the Riemann problem centred at two different points for a system of equations modelling propagation of elastic waves. The system consists of two equations for $(u,\sigma)$, where, u is the velocity and $\sigma$ is the stress and is strictly hyperbolic and nonconservative. Study of interaction of waves is one of the most important steps in the construction of global solution with initial data in the space of functions of bounded variation using approximation procedure like the Glimm's scheme. This amounts to constructing a solution with initial data consisting of three states $(u_L, \sigma_L), (u_m, \sigma_m),$ and $(u_R, \sigma_R)$. Usually this analysis is done for the states which are in a small neighbourhood of a fixed state. Here we get explicit formula for the solution of the system when the data lies in the level sets of Riemann invariants. The speciality of the work is that we donot assume smallness conditions on the initial data.