A hidden Markov model for the dynamics of diffusing dissipative solitons

We investigate the dynamics of dissipative solitons in the cubic-quintic complex Ginzburg-Landau equation in one spatial dimension for different values of the bifurcation parameter mu. We consider a certain range of the parameter mu where dissipative solitons show explosions, i.e. transient enlargements of the soliton that lead to spatial shifts if the explosions are asymmetric. We find that depending on the parameter mu, the arising sequence of spatial shifts can be modeled by a simple anti-persistent random walk or by a more complicated hidden Markov model. We show with the help of exact analytical calculations that these models are able to reproduce several statistics of the soliton motion such as the distribution of spatial shifts, the correlation of spatial shifts, and the distribution of zig-zag streaks.