Wave Interactions in Nonlinear Strings

Abstract

We study the interactions of nonlinear waves in a system for nonlinear elastic strings. The system can be written where and, so we have a 6 x 6 system of hyperbolic conservation laws in one space dimension. This is a model for the more general system for elasticity, of which it can be considered a special case, with extra symmetries. There is a full set of eigenvectors provided the tension T and its derivative T’ don’t vanish, even when the eigenvalues coincide. The system is symmetric hyperbolic, but strict hyperbolicity fails in two ways. First, there are only four distinct eigenvalues, two of which have multiplicity two, and thus degenerate. Rather than a wave curve as in the classical case, a (compact) wave surface, namely a sphere, is associated with each of the degenerate families. The second degeneracy occurs because the wavespeeds cross, and so cannot be ordered.