Class of compound dissipative solitons as a result of collisions in one and two spatial dimensions

Collisions of non-explosive dissipative solitons can induce explosions

We investigate the interaction of stationary and oscillatory dissipative solitons in the framework of two coupled cubic-quintic complex Ginzburg-Landau equation for counter-propagating waves. We analyze the case of a stabilizing as well as a destabilizing cubic cross-coupling between the counter-propagating dissipative solitons. The three types of interacting localized solutions investigated are stationary, oscillatory with one frequency, and oscillatory with two frequencies. We show that there is a large number of different outcomes as a result of these collisions including stationary as well as oscillatory bound states and compound states with one and two frequencies. The two most remarkable results are (a) the occurrence of bound states and compound states of exploding dissipative solitons as outcome of the collisions of stationary and oscillatory pulses; and (b) spatiotemporal disorder due to the creation, interaction, and annihilation of dissipative solitons for colliding oscillatory dissipative solitons as initial conditions.

On the influence of additive and multiplicative noise on holes in dissipative systems

We investigate the influence of noise on deterministically stable holes in the cubic-quintic complex Ginzburg-Landau equation. Inspired by experimental possibilities, we specifically study two types of noise: additive noise delta-correlated in space and spatially homogeneous multiplicative noise on the formation of π-holes and 2π-holes. Our results include the following main features. For large enough additive noise, we always find a transition to the noisy version of the spatially homogeneous finite amplitude solution, while for sufficiently large multiplicative noise, a collapse occurs to the zero amplitude solution. The latter type of behavior, while unexpected deterministically, can be traced back to a characteristic feature of multiplicative noise; the zero solution acts as the analogue of an absorbing boundary: once trapped at zero, the system cannot escape. For 2π-holes, which exist deterministically over a fairly small range of values of subcriticality, one can induce a transition to a π-hole (for additive noise) or to a noise-sustained pulse (for multiplicative noise). This observation opens the possibility of noise-induced switching back and forth from and to 2π-holes.

Multiplicative noise can lead to the collapse of dissipative solitons

We investigate the influence of spatially homogeneous multiplicative noise on the formation of localized patterns in the framework of the cubic-quintic complex Ginzburg-Landau equation. We find that for sufficiently large multiplicative noise the formation of stationary and temporally periodic dissipative solitons is suppressed. This result is characterized by a linear relation between the bifurcation parameter and the noise amplitude required for suppression. For the regime associated with exploding dissipative solitons we find a reduction in the number of explosions for larger noise strength as well as a conversion to other types of dissipative solitons or to filling-in and eventually a collapse to the zero solution.

Non-unique results of collisions of quasi-one-dimensional dissipative solitons

We investigate collisions of quasi-one-dimensional dissipative solitons (DSs) for a large class of initial conditions, which are not temporally asymptotic quasi-one-dimensional DSs. For the case of sufficiently small approach velocity and sufficiently large values of the dissipative cross-coupling between the counter-propagating DSs, we find non-unique results for the outcome of collisions. We demonstrate that these non-unique results are intrinsically related to a modulation instability along the crest of the quasi-one-dimensional objects. As a model, we use coupled cubic-quintic complex Ginzburg-Landau equations. Among the final results found are stationary and oscillatory compound states as well as more complex assemblies consisting of quasi-one-dimensional and localized states. We analyse to what extent the final results can be described by the solutions of one cubic-quintic complex Ginzburg-Landau equation with effective parameters.

Dissipative solitons and backfiring in the electrooxidation of CO on Pt

Collisions of excitation pulses in dissipative systems lead usually to their annihilation. In this paper, we report electrochemical experiments exhibiting more complex pulse interaction with collision survival and pulse splitting, phenomena that have rarely been observed experimentally and are only poorly understood theoretically. Using spatially resolved in-situ Fourier transform infrared spectroscopy (FTIR) in the attenuated total reflection configuration, we monitored reaction pulses during the electrochemical oxidation of CO on Pt thin film electrodes in a flow cell. The system forms quasi-1d pulses that align parallel to the flow and propagate perpendicular to it. The pulses split once in a while, generating a second solitary wave in the backward moving direction. Upon collision, the waves penetrate each other in a soliton-like manner. These unusual pulse dynamics could be reproduced with a 3-component reaction-diffusion-migration model with two inhibitor species, one of them exhibiting a long-range spatial coupling. The simulations shed light on existence criteria of such dissipative solitons.

Noisy localized structures induced by large noise

We investigate the influence of large noise on the formation of localized patterns in the framework of the cubic-quintic complex Ginzburg-Landau equation. The interaction of localization and noise can lead to filling in or noisy localized structures for fixed noise strength. To focus on the interaction between noise and localization we cover a region in parameter space, in particular, subcriticality, for which stationary stable deterministic pulses do not exist. Possible experimental tests of the work presented for autocatalytic chemical reactions and bioinspired systems are outlined.