Linear and orbital stability analysis for solitary-wave solutions of variable-coefficient scalar-field equations

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