Phase Plane Behavior of Solitary Waves in Nonlinear Layered Media

The one-dimensional elastic wave equations for compressional waves have the form ∈_t(x,t)−u_x(x,t) = 0 (ρ(x)u(x,t))_t−σ(∈(x,t),x)_x = 0 (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.