Nonstrictly hyperbolic waves in elasticity

We consider a system modeling the dynamics of a nonlinear elastic string. This 6×6 system is nonstrictly hyperbolic, having two families with multiplicity two. Moreover, the distinct wavespeeds cross, giving a further degeneracy. It is essential to consider the multiplicity of eigenvalues when checking entropy conditions to ensure uniqueness. Because the nonlinearity appears through a single scalar function T(u), the Riemann problem can be analyzed in detail by a construction analogous to Oleinikʹs. We solve the Riemann problem for this system with large data, and give a qualitative description of the interactions of nonlinear waves of arbitrary strength.