Soliton diffusion on chains of coupled nonlinear oscillators

We study the diffusivity of solitons in noisy-coupled nonlinear oscillators. We consider an anharmonic atomic chain and a network of coupled RLC circuits with nonlinear capacitance and linear inductances and resistances. The solitons propagate in the presence of either thermal noise and/or shot noise. We use a Langevin formalism in order to model the noisy systems, for that reason noise and damping are added to the discrete equations of motion. In the long-wave approximation both systems can be described by identical noisy and damped Korteweg–de Vries (KdV) equations. By using the results of the well-known adiabatic perturbation theory of the forced KdV equation we derive ordinary differential equations (ODEs) for the relevant variables of the soliton, namely position and inverse of the width. We solve these ODEs by using standard perturbation theory to obtain analytical expressions of the variance and average of the soliton position. We perform Langevin dynamics simulations of the full discrete systems which confirm our analytical results, namely superdiffusivity of the solitons depending on the initial velocity.