Long-range effects on superdiffusive solitons in anharmonic chains

Abstract

Studies on thermal diffusion of lattice solitons in Fermi-Pasta-Ulam (FPU)-like lattices were recently generalized to the case of dispersive long-range interactions (LRI) of the Kac-Baker form. The position variance of the soliton shows a stronger than linear time-dependence (superdiffusion) as found earlier for lattice solitons on FPU chains with nearest neighbour interactions (NNI). In contrast to the NNI case where the position variance at moderate soliton velocities has a considerable linear time-dependence (normal diffusion), the solitons with LRI are dominated by a superdiffusive mechanism where the position variance mainly depends quadratic and cubic on time. Since the superdiffusion seems to be generic for nontopological solitons, we want to illuminate the role of the soliton shape on the superdiffusive mechanism. Therefore, we concentrate on a FPU-like lattice with a certain class of power-law long-range interactions where the solitons have algebraic tails instead of exponential tails in the case of FPU-type interactions (with or without Kac-Baker LRI). A collective variable (CV) approach in the continuum approximation of the system leads to stochastic integro-differential equations which can be reduced to Langevin-type equations for the CV position and width. We are able to derive an analytical result for the soliton diffusion which agrees well with the simulations of the discrete system. Despite of structurally similar Langevin systems for the two soliton types, the algebraic solitons reach the superdiffusive long-time limit with a characteristic $t^{1.5}$ time-dependence much faster than exponential solitons. The soliton shape determines the diffusion constant in the long-time limit that is approximately a factor of $\pi$ smaller for algebraic solitons.

7 figures

References in corpus (7)

• Breathers on lattices with long range interaction
• Energy Relaxation in Nonlinear One-Dimensional Lattices
• Optical creation of vibrational intrinsic localized modes in anharmonic lattices with realistic interatomic potentials
• Thermal diffusion of supersonic solitons in an anharmonic chain of atoms
• Solitons in anharmonic chains with ultra-long-range interatomic interactions
• Multi-component structure of nonlinear excitations in systems with length-scale competition
• Thermal diffusion of solitons on anharmonic chains with long-range coupling