Nonlinear Fractional Dynamics of Lattice with Long-Range Interaction

A unified approach has been developed to study nonlinear dynamics of a 1D lattice of particles with long-range power-law interaction. Classical case is treated in the framework of the well-known FrenkelKontorova chain model. Qunatum dynamics is considered follow to Davydov’s approach to 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 sineGordon, nonlinear Schrödinger, and Hilbert-Schrödinger equations have been found. There exists a critical value of the powers of the long-range potential. Below the critical value (s < 3, s 6= 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 quantum lattice propagator has been studied. We have shown that the quantum exciton propagator exhibits transition from the well-known Gaussian-like behavior to power-law decay due to long-range interaction. Link between 1D quantum lattice dynamics in the imanaginary time domain and random walk model has been discussed. PACS number(s): 03.65.-w, 03.65. Db, 05.30.-d, 05.40. Fb