Marcel G. Clerc

Localized structures in dissipative media: from optics to plant ecology

Localized structures (LSs) in dissipative media appear in various fields of natural science such as biology, chemistry, plant ecology, optics and laser physics. The proposal for this Theme Issue was to gather specialists from various fields of nonlinear science towards a cross-fertilization among active areas of research. This is a cross-disciplinary area of research dominated by nonlinear optics due to potential applications for all-optical control of light, optical storage and information processing. This Theme Issue contains contributions from 18 active groups involved in the LS field and have all made significant contributions in recent years.

Bouncing localized structures in a liquid-crystal light-valve experiment.

In a liquid crystal light-valve experiment, we report solitary localized structures appearing outside the bistability range and displaying a behavior of single independent cells. The transition from an extended pattern to solitary states is characterized both experimentally and numerically.

Additive Noise Induces Front Propagation

The propagation of a front connecting a stable homogeneous state with a stable periodic state in the presence of additive noise is studied. The mean velocity was computed both numerically and analitically. The numerics are in good agreement with the analitical prediction.

Spatiotemporal Chaos Induces Extreme Events in an Extended Microcavity Laser

Extreme events such as rogue waves in optics and fluids are often associated with the merging dynamics of coherent structures. We present experimental and numerical results on the physics of extreme event appearance in a spatially extended semiconductor microcavity laser with an intracavity saturable absorber. This system can display deterministic irregular dynamics only, thanks to spatial coupling through diffraction of light. We have identified parameter regions where extreme events are encountered and established the origin of this dynamics in the emergence of deterministic spatiotemporal chaos, through the correspondence between the proportion of extreme events and the dimension of the strange attractor.

Localized patterns and hole solutions in one-dimensional extended systems

The existence, stability properties, and bifurcation diagrams of localized patterns and hole solutions in one-dimensional extended systems is studied from the point of view of front interactions. An adequate envelope equation is derived from a prototype model that exhibits these particle-type solutions. This equation allow us to obtain an analytical expression for the front interaction, which is in good agreement with numerical simulations.

Strong interaction between plants induces circular barren patches: fairy circles.

Fairy circles consist of isolated or randomly distributed circular areas devoid of any vegetation. They are observed in vast territories in southern Angola, Namibia and South Africa. We report on the formation of fairy circles, and we interpret them as localized structures with a varying plateau size as a function of the aridity. Their stabilization mechanism is attributed to a combined influence of the bistability between the bare state and the uniformly vegetation state, and Lorentzian-like non-local coupling that models the competition between plants. We show how a circular shape is formed, and how the aridity level influences the size of fairy circles. Finally, we show that the proposed mechanism is model-independent.

NONVARIATIONAL ISING–BLOCH TRANSITION IN PARAMETRICALLY DRIVEN SYSTEMS

Transition from motionless to moving domain walls connecting two uniform oscillatory equivalent states in both a magnetic wire forced with a transversal oscillating magnetic field and a parametrically driven damped pendula chain are studied. These domain walls are not contained in the conventional approach to these systems — parametrically driven damped nonlinear Schrodinger equation. By adding in this model higher order terms, we are able to explain these solutions and the transition between resting and moving walls. Based on amended amplitude equation, we deduced a set of ordinary differential equations which describes the nonvariational Ising–Bloch transition in unified manner.

Localized states in bistable pattern forming systems

We present a unifying description close to a spatial bifurcation of localized states, appearing as large amplitude peaks nucleating over a pattern of lower amplitude. Localized states are pinned over a lattice spontaneously generated by the system itself. We show that the phenomenon is generic and requires only the coexistence of two spatially periodic states. At the onset of the spatial bifurcation, a forced amplitude equation is derived for the critical modes, which accounts for the appearance of localized peaks.

Chimera-type states induced by local coupling.

Coupled oscillators can exhibit complex self-organization behavior such as phase turbulence, spatiotemporal intermittency, and chimera states. The latter corresponds to a coexistence of coherent and incoherent states apparently promoted by nonlocal or global coupling. Here we investigate the existence, stability properties, and bifurcation diagram of chimera-type states in a system with local coupling without different time scales. Based on a model of a chain of nonlinear oscillators coupled to adjacent neighbors, we identify the required attributes to observe these states: local coupling and bistability between a stationary and an oscillatory state close to a homoclinic bifurcation. The local coupling prevents the incoherent state from invading the coherent one, allowing concurrently the existence of a family of chimera states, which are organized by a homoclinic snaking bifurcation diagram.

Continuous description of lattice discreteness effects in front propagation

Models describing microscopic or mesoscopic phenomena in physics are inherently discrete, where the lattice spacing between fundamental components, such as in the case of atomic sites, is a fundamental physical parameter. The effect of spatial discreteness over front propagation phenomenon in an overdamped one-dimensional periodic lattice is studied. We show here that the study of front propagation leads in a discrete description to different conclusions that in the case of its, respectively, continuous description, and also that the results of the discrete model, can be inferred by effective continuous equations with a supplementary spatially periodic term that we have denominated Peierls–Nabarro drift, which describes the bifurcation diagram of the front speed, the appearance of particle-type solutions and their snaking bifurcation diagram. Numerical simulations of the discrete equation show quite good agreement with the phenomenological description.