Three excitons states in nonlinear saturation α -helix protein

In the present work, the effects of nonlinear saturations on the dynamics of three excitons in the protein chain are investigated. Extending the Davydov Hamiltonian to three excitons, it is shown that the system reduces to three-coupled nonlinear Schrödinger equations with three additional saturation terms. Then, it is proved that nonlinear saturations may be used to stabilize the system under modulational instability. Besides, many solutions of the three-coupled nonlinear Schrödinger equations with saturation terms are constructed and classified in three families, i.e., hyperbolic functions which includes bright, kink and dark solitary waves, triangular periodic and generalized Jacobi elliptic functions. The analytical predictions are confirmed by numerical simulations with a good accuracy. It appears that nonlinear saturations profoundly affect the dynamics of the solutions. These solutions may be used to explain the energy transfer and flow through proteins chains.