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

Abstract

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.

Chaotic Diffusion of Dissipative Solitons: From Anti-Persistent Random Walk to Hidden Markov Models

“…When the value of the bifurcation parameter µ becomes even smaller, this weight goes to zero and only the second APRW generated by the states 3 and 4 of the original HMM remains. This APRW can be identified with the one that was found and studied in detail in a previous publication [39] for µ = −0.18.…”

“…In a previous article [39], we investigated the soliton dynamics for µ = −0.18, where according to Fig. 2 (c only two sharp peaks are visible in the distribution of the spatial shifts.…”

“…In Sec. IV, we give an overview of our findings from previous publications [39,40] before we present new complementary results in Sec. V. Our findings are discussed in Sec.…”

“…In previous publications, we identified the optimal model parameters Λ * for two cases of the CGL. In [40], we identified Λ * (µ = −0.10), which corresponds to a 4-state HMM as introduced above, whereas in [39], we found that the diffusive behavior of the solitons for µ = −0.18 is well described by an Anti-Persistent Random Walk (APRW), whose generator is basically a 2-state Markov process, i.e., it is not an HMM, or in some sense a degenerate case of an HMM. Now, one expects that Λ * (µ) is continuously varying with µ.…”

“…In one spatial dimension, this is just a scalar quantity, which can be regarded as a projection from infinite-dimensional state space onto the spatial coordinate. Recently, we showed for one specific case in one dimension that the center of mass can be described phenomenologically as an Anti-Persistent Random Walk [39]. In general, however, even in one spatial dimension, the situation is much more complex.…”