OSCILLATORY INSTABILITY OF SOLITARY WAVES IN A CONTINUUM MODEL OF LATTICE-VIBRATIONS

We study the stability of solitary waves of two coupled Boussinesq equ ations which model weakly nonlinear vibrations in a cubic lattice. A H amiltonian formulation is presented. Known variational methods are obs erved to be incapable of establishing stability or detecting instabili ty. Instead, the problem is linearized and studied using the Evans fun ction, an analytic function whose zeros, when in the right half plane, correspond to discrete unstable eigenvalues. It is proved that if the re is a linear exponential instability, then at transition a pair of c omplex conjugate eigenvalues emerges into the right half plane. The Ev ans 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 Christians en, Lomdahl and Mute (1990 Nonlinearity 4 477), which were drawn from numerically computed solutions of the initial-value problem. The insta bility mechanism in this system differs from that typical in finite-di mensional 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 lef t half plane, that cross the continuous spectrum and emerge as unstabl e eigenvalues.