  We consider linear instability of solitary waves of several classes of
dispersive long wave models. They include generalizations of KDV, BBM,
regularized Boussinesq equations, with general dispersive operators and
nonlinear terms. We obtain criteria for the existence of exponentially growing
solutions to the linearized problem. The novelty is that we dealt with models
with nonlocal dispersive terms, for which the spectra problem is out of reach
by the Evans function technique. For the proof, we reduce the linearized
problem to study a family of nonlocal operators, which are closely related to
properties of solitary waves. A continuation argument with a moving kernel
formula are used to find the instability criteria. Recently, these techniques
have also been extended to study instability of periodic waves and to the full
water wave problem.
