Resonant two-wave interaction in the Davydov model

Abstract

The Davydov model for exciton–phonon coupling in hydrogen-bonded molecular chains is reconsidered in the context of two-wave resonant interaction. By applying a semi-discrete slowly varying envelope approximation, when the physical problem is that of the long-distance evolution of an input finite duration excitonic pulse, we derive an integrable limit model which preserves the coupling nature of the process. The spectral transform is constructed with emphasis on the complete characterization of the spectral data. As an application, the localized one-soliton solution is explicitly constructed. Then by using Darboux–B¨ acklund transformations, a non-local (or topological) one-soliton solution is also derived. As a consequence, the system possess two different soliton solutions where the phonon component is a localized pulse, but where the exciton wave is either localized (bell shape) or topological (kink shape). The resulting approximate soliton solutions of the Davydov model in the resonant regime are subsonic in the localized case and supersonic in the topological case. Finally, by expressing the Bianchi superposition theorem, a nonlinear superposition formula is derived allowing for explicit two-soliton solution.