  We study general semilinear scalar-field equations on the real line with
variable coefficients in the linear terms. These coefficients are uniformly
small, but slowly decaying, perturbations of a constant-coefficient operator.
We are motivated by the question of how these perturbations of the equation may
change the stability properties of kink solutions (one-dimensional topological
solitons). We prove existence of a stationary kink solution in our setting, and
perform a detailed spectral analysis of the corresponding linearized operator,
based on perturbing the linearized operator around the constant-coefficient
kink. We derive a formula that allows us to check whether a discrete eigenvalue
emerges from the essential spectrum under this perturbation. Known examples
suggest that this extra eigenvalue may have an important influence on the
long-time dynamics in a neighborhood of the kink. We also establish orbital
stability of solitary-wave solutions in the variable-coefficient regime,
despite the possible presence of negative eigenvalues in the linearization.
