Coupled fractional nonlinear differential equations and exact Jacobian elliptic solutions for exciton-phonon dynamics

An improved quantum model for exciton-phonon dynamics in an α-helix is investigated taking into account the interspine coupling and the influence of power-law long-range exciton-exciton interactions. Having constructed the model Hamiltonian, we derive the lattice equations and employ the Fourier transforms to go in continuum space showing that the long-range interactions (LRI) lead to a nonlocal integral term in the equations of motion. Indeed, the non-locality originating from the LRI results in the dynamic equations with space derivatives of fractional order. New theoretical frameworks are derived, such that: fractional generalization of coupled Zakharov equations, coupled nonlinear fractional Schrödinger equations, coupled fractional Ginzburg-Landau equations, coupled Hilbert-Zakharov equations, coupled nonlinear Hilbert-Ginzburg-Landau equations, coupled nonlinear Schrödinger equations and coupled nonlinear Hilbert-Schrödinger equations. Through the F-expansion method, we derive a set of exact Jacobian solutions of coupled nonlinear Schrödinger equations. These solutions include Jacobian periodic solutions as well as bright and dark soliton which are important in the process of energy transport in the molecule. We also discuss of the impact of LRI on the energy transport in the molecule.