Nonlinear fractional dynamics on a lattice with long range interactions

A unified approach has been developed to study nonlinear dynamics of a 1D lattice of particles with long-range power-law interaction. A classical case is treated in the framework of the generalization of the well-known Frenkel–Kontorova chain model for the non-nearest interactions. Quantum dynamics is considered following Davydovʹs approach for molecular excitons. In the continuum limit the problem is reduced to dynamical equations with fractional derivatives resulting from the fractional power of the long-range interaction. Fractional generalizations of the sine-Gordon, nonlinear Schrödinger, and Hilbert–Schrödinger equations have been found. There exists a critical value of the power s of the long-range potential. Below the critical value (s<3, s≠1, 2) we obtain equations with fractional derivatives while for s 3 we have the well-known nonlinear dynamical equations with space derivatives of integer order. Long-range interaction impact on the quantum lattice propagator has been studied. We have shown that the quantum exciton propagator exhibits transition from the well-known Gaussian-like behavior to a power-law decay due to the long-range interaction. A link between 1D quantum lattice dynamics in the imaginary time domain and a random walk model has been discussed.