Integrable systems related to Su–Schrieffer–Heeger lattices

We present a novel approach to nonlinear lattices with fermions based on the exact solutions for the equations of motion, where fermions and bosons are treated as Grassmannian anticommuting and commuting classical fields. It is shown that the Su–Schrieffer–Heeger equations of motion can be solved exactly in continuum limit using Hirota bilinear formalism and bright and dark solitons appear for phononic and fermionic fields. The bosonic soliton has a tail due to the presence of fermions and the fermionic density has a strongly localised shape. Moreover, applying the multiple scales method, the emergent Grassmannian amplitude nonlinear equation is the complex modified Korteweg de Vries and at the resonance the fermionic version of Yajima–Oikawa system. As an interesting offshoot this system turns out to be a new completely integrable one having N-dark soliton solution.