Stability of multi-hump localized solutions in the Holstein model for linear acoustic and soft nonlinear optical interactions

We consider the energy/electron transport problem along a lattice holding linear acoustic and soft cubic non-dispersive interactions in the long wave limit for Holstein's approach. It is found that the model equations, consisting of the coupling between the linear Schrödinger and a nonlinear Klein–Gordon equations, support one- and two-hump solutions of the first and second kind. We derive these multi-humped solutions by using a compatible reduction of the equations of motion and by means of an adaptation of the direct method, both in the traveling frame. Remarkably, the solution profiles for the mechanical deformation recall the one and two soliton solutions of the Korteweg–de Vries equation. Then, based on the Fourier collocation method, we perform a numerical linear stability analysis of these solutions at the steady state to find the parameters regime for stability. Additionally, we are able to implement the Vakhitov–Kolokolov criterion to get the bifurcation value for the one-hump solution both of the first and second kind. We show that the two-hump solutions are linearly unstable.