We study a type of `eigenvalue' problem for systems of linear ordinary differential equations with asymptotically constant coefficients by using the analytic function D($\lambda $) introduced by J. W. Evans (1975) in his study of the stability of nerve impulses. We develop a general theory of D($\lambda $) that clarifies the role of the essential spectrum in applications. New formulae for derivatives of D($\lambda $) 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.