  Frequently encountered in nature, internal solitary waves in stratified
fluids are well-observed and well-studied from the experimental, the
theoretical, and the numerical perspective. From the mathematical point of
view, these waves are exact solutions of the 2D Euler equations for
incompressible, inviscid fluids. Contrasting with a rich theory for their
existence and the development of methods for computing these waves, their
stability analysis has hardly received attention at a rigorous mathematical
level.
  This paper proposes a new approach to the investigation of stability of
internal solitary waves in a continuously stratified fluidic medium and carries
out the following four steps of this approach: (I) to formulate the eigenvalue
problem as an infinite-dimensional spatial-dynamical system, (II) to introduce
finite-dimensional truncations of the spatial-dynamics description, (III) to
demonstrate that each truncation, of any order, permits a well-defined Evans
function, (IV) to prove absence of small zeros of the Evans function in the
small-amplitude limit. The latter notably implies the low-frequency spectral
stability of small-amplitude internal solitary waves to arbitrarily high
truncation order.
