Phase Plane Behavior of Solitary Waves in Nonlinear Layered Media

Abstract

The one-dimensional elastic wave equations for compressional waves have the form

$$ \begin{array}{*{20}{c}} \hfill {{{ \in }_{t}}(x,t) - {{u}_{x}}(x,t) = 0} \\ \hfill {{{{(\rho (x)u(x,t))}}_{t}} - \sigma {{{( \in (x,t),x)}}_{x}} = 0} \\ \end{array} $$

((1))

where ε(x, t) is the strain and u(x, t) the velocity. We consider a heterogeneous material with the density specified by ρ(x) and a nonlinear constitutive relation for the stress given by a function σ(∈, x) that also varies explicitly with x. This is a hyperbolic system of conservation laws with a spatially-varying flux function, q _t + f(q, x) _x = 0.