Non-uniqueness for the nonlinear dynamical Lamé system

We consider the Cauchy problem for the nonlinear dynamical Lamé system with double wave speeds in a $d$-dimensional $(d=2,3)$ periodic domain. Moreover, the equations can be transformed into a linearly degenerate hyperbolic system. We could construct infinitely many continuous solutions in $C^{1,\alpha }$ emanating from the same small initial data for $\alpha <\frac{1}{60}$. The proof relies on the convex integration scheme. We construct a new class of building blocks with compression structure by using the double wave speeds characteristic of the equations.