A new kind of chaotic diffusion: Anti-persistent random walks of explosive dissipative solitons

The solitons that exist in nonlinear dissipative media have properties very different from the ones that exist in conservative media and are modeled by the nonlinear Schrödinger equation. One of the surprising behaviors of dissipative solitons is the occurrence of explosions: sudden transient enlargements of a soliton, which as a result induce spatial shifts. In this work using the complex Ginzburg-Landau equation in one dimension, we address the long-time statistics of these apparently random shifts. We show that the motion of a soliton can be described as an anti-persistent random walk with a corresponding oscillatory decay of the velocity correlation function. We derive two simple statistical models, one in discrete and one in continuous time, which explain the observed behavior. Our statistical analysis benchmarks a future microscopic theory of the origin of this new kind of chaotic diffusion.