Eigenvalues, and Instabilities of Solitary Waves

We study a type of ‘eigenvalue’ problem for systems of linear ordinary differential equations with asymptotically constant coefficients by using the analytic function D(λ) introduced by J. W. Evans (1975) in his study of the stability of nerve impulses. We develop a general theory of D(λ) that clarifies the role of the essential spectrum in applications. New formulae for derivatives of D(λ) are used to study linear exponential instabilities of solitary waves for generalizations of: (1) the Korteweg-de Vries equation (KdV); (2) the Benjamin-Bona-Mahoney equation (BBM); and (3) the regularized Boussinesq equation. A pair of real eigenvalues exists, indicating a non-oscillatory instability, when the ‘momentum’ of the wave (a time-invariant functional associated with the hamiltonian structure of the equation) is a decreasing function of wave speed. Also we explain the mechanism of the transition to instability. Unexpectedly, these transitions are unlike typical transitions to instability in finite-dimensional hamiltonian systems. Instead they can be understood in terms of the motion of poles of the resolvent formula extended to a multi-sheeted Riemann surface. Finally, for a generalization of the KdV-Burgers equation (a model for bores), we show that a conjectured transition to instability does not involve real eigenvalues emerging from the origin, suggesting an oscillatory type of instability.