  This paper studies the linear stability problem for solitary wave solutions
of Hamiltonian PDEs. The linear stability problem is formulated in terms of the
Evans function, a complex analytic function denoted by $D(\lambda)$, where
$\lambda$ is the spectral parameter. The main result is the introduction of a
new factor, denoted $\Pi$, in the Pego & Weinstein (1992) derivative formula \[
D''(0) = \chi \Pi \frac{dI}{dc}\,, \] where $I$ is the momentum of the solitary
wave and $c$ is the speed. Moreover this factor turns out to be related to
transversality of the solitary wave, modelled as a homoclinic orbit: the
homoclinic orbit is transversely constructed if and only if $\Pi\neq 0$. The
sign of $\Pi$ is a symplectic invariant, an intrinsic property of the solitary
wave, and is a key new factor affecting the linear stability. The factor $\chi$
was already introduced by Bridges & Derks (1999) and is based on the
asymptotics of the solitary wave. A supporting result is the introduction of a
new abstract class of Hamiltonian PDEs built on a nonlinear Dirac-type
equation, which model a wide range of PDEs in applications. Examples where the
theory applies, other than Dirac operators, are the coupled mode equation in
fluid mechanics and optics, the massive Thirring model, and coupled nonlinear
wave equations. A calculation of $D''(0)$ for solitary wave solutions of the
latter class is included to illustrate the theory.
