Effective Lax-entropy condition on two-dimensional layered media

Here we study the conditions to have an stable shock in two-dimensional layered medium. We consider three situations:

• z-dispersive media
• c-dispersive media
• cz-dispersive media

Based on theory of nonlinear hyperbolic equations, it is known that shocks occur when characteristics of different families collide. Consider, for instance, a right going shock. Then, the Lax-entropy condition impose a velocity on an stable shock to be:

\(\begin{align} \lambda_l>S>\lambda_r, \end{align} \)

where \(\lambda_l\) and \(\lambda_r\) are the positive eigenvalues at the left and right state of the shock and \(S\) is the shock speed. This condition implies that right-going characteristics will collide (on the x-t plane).

It is known that waves moving on periodic media may exhibit shock formation depending on the strength (or energy) of the initial condition and the material parameters. For a fixed initial condition, as we consider media with more effective (\(c\)-, \(z\)- or \(cz\)-) dispersion, shock formation becomes harder. On the other hand, if the material parameters are fixed, we can control shock formation by considering initial conditions with less or more energy; as we consider initial conditions with more energy, more chance the solution develop shocks.

In this section we try to understand when shocks will develop from smooth initial conditions and when initial shocks will persist. This is done by obtaining an effective Lax-entropy condition.