Oscillatory instability of solitary waves in a continuum model of lattice vibrations

Abstract

We study the stability of solitary waves of two coupled Boussinesq equations which model weakly nonlinear vibrations in a cubic lattice. A Hamiltonian formulation is presented. Known variational methods are observed to be incapable of establishing stability or detecting instability. Instead, the problem is linearized and studied using the Evans function, an analytic function whose zeros, when in the right half plane, correspond to discrete unstable eigenvalues. It is proved that if there is a linear exponential instability, then at transition a pair of complex conjugate eigenvalues emerges into the right half plane. The Evans function is computed numerically and we observe complex conjugate pairs of zeros crossing into the right half plane. The first pair that crosses does so in close agreement with the conclusions of Christiansen, Lomdahl and Muto (1990 Nonlinearity 4 477), which were drawn from numerically computed solutions of the initial-value problem. The instability mechanism in this system differs from that typical in finite-dimensional Hamiltonian systems, where transition to instability occurs via collisions of imaginary eigenvalues. Here, the transition involves resonance poles, which are the zeros of the Evans function in the left half plane, that cross the continuous spectrum and emerge as unstable eigenvalues.