# D. Walgraef's research while affiliated with University of the Balearic Islands and other places

## Publications (167)

Preprint
Full-text available
Nonlinear models for pattern evolution by ion beam sputtering on a material surface present an ongoing opportunity for new numerical simulations. A numerical analysis of the evolution of preexisting patterns is proposed to investigate surface dynamics, based on a 2D anisotropic damped Kuramoto-Sivashinsky equation, with periodic boundary conditions...
Article
Full-text available
Nonlinear models for pattern evolution by ion beam sputtering on a material surface present an ongoing opportunity for new numerical simulations. A numerical analysis of the evolution of preexisting patterns is proposed to investigate surface dynamics, based on a 2D anisotropic damped Kuramoto-Sivashinsky equation, with periodic boundary conditions...
Article
Full-text available
We present a continuum model of ion-induced surface patterning. The model incorporates the atomic processes of sputtering, re-deposition and surface diffusion, and is shown to display the generic features of the damped Kuramoto-Sivashinsky (KS) equation of non-linear dynamics. Linear and non-linear stability analyses of the evolution equation give...
Article
Spatially extended systems can support local transient excitations in which just a part of the system is excited. The mechanisms reported so far are local excitability and excitation of a localized structure. Here we introduce an alternative mechanism based on the coexistence of two homogeneous stable states and spatial coupling. We show the existe...
Article
We derive a generic model for the interaction of domain walls close to a nonequilibrium-Bloch transition. The universal scenario predicted by the model includes stationary Ising and Bloch localized structures (dissipative solitons), as well as drifting and oscillating Bloch structures. Our theory also explains the behavior of Bloch walls during a c...
Article
A dynamical model of the Swift-Hohenberg type is proposed to describe the formation of twelvefold quasipattern as observed, for instance, in optical systems. The model incorporates the general mechanisms leading to quasipattern formation and does not need external forcing to generate them. Besides quadratic nonlinearities, the model takes into acco...
Article
When thin films are grown on a substrate by chemical vapor deposition, the evolution of the first deposited layers may be described, on mesoscopic scales, by dynamical models of the reaction-diffusion type. For monatomic layers, such models describe the evolution of atomic coverage due to the combined effect of reaction terms representing adsorptio...
Article
When thin films are grown on a substrate by chemical vapor deposition, the evolution of the first deposited layers may be described, on mesoscopic scales, by dynamical models of the reaction-diffusion type. For monoatomic layers, such models describe the evolution of atomic coverage due to the combined effect of reaction terms representing adsorpti...
Article
It is shown how the combination of atomic deposition and nonlinear diffusion may lead, below a critical temperature, to the growth of nonuniform layers on a substrate. The dynamics of such a system is of the Cahn-Hilliard type, supplemented by reaction terms representing adsorption-desorption processes. The instability of uniform layers leads to th...
Article
Deformation-induced material instabilities may be of elastic (buckling, martensitic transformations) or plastic (necking/shear banding, dislocation patterning) type. In plasticity, the emergence of material instabilities is mainly associated with the properties of the underlying microstructure such as the motion, interaction and production/annihila...
Article
Strain localization and dislocation pattern formation are typical features of plastic deformation in metals and alloys. Glide and climb dislocation motion along with accompanying production/ annihilation processes of dislocations lead to the occurrence of instabilities of initially uniform dislocation distributions. These instabilities result into...
Article
Strain localization and dislocation pattern formation are typical features of plastic deformation in metals and alloys. Glide and climb dislocation motion along with accompanying production/annihilation processes of dislocations lead to the occurrence of instabilities of initially uniform dislocation distributions. These instabilities result into t...
Article
Full-text available
In Part I, the W-A model for dislocation patterning was revisited by elaborating on gradient terms associated with the dominant dislocation dynamics mechanisms. However, the phenomenological coefficients entering in this dynamics were left to be determined by experiment or discrete dislocation mechanisms arguments, and stochastic aspects were negle...
Article
The effect of external fields on spatio-temporal Hopf bifurcations is investigated. Left- and right-travelling waves are linearly coupled by uniform oscillations or steady spatial modulations, provided the frequency of the oscillations or the wave number of the modulations are close to two times the critical ones. Hence, a spatially uniform forcing...
Article
The effect of two-dimensional spatial modulations on spatio-temporal Hopf bifurcations is investigated. In particular, in the case of a spatial forcing corresponding to triangular or hexagonal planforms, a strong resonance between triplets of travelling waves may occur, provided that the wavelength of the forcing is nearly equal to one-third of the...
Article
This paper reviews some of the main mechanical properties of carbon nanotubes (CNT), including nonlinear elastic instabilities and buckling, as well as defect generation, dynamics and plasticity. A multiscale modeling approach of CNT dynamics is proposed. It considers CNT as cylindrical shells and describes their dynamics through the corresponding...
Chapter
IntroductionCritical Phenomena in Multiple Steady-State SystemsPhase Transitions to Nonuniform Steady StatesHard-Mode Transition to Chemical Oscillations
Article
Strain localization and dislocation pattern formation are typical features of plastic deformation in metals and alloys. Glide and climb dislocation motion, along with accompanying production/annihilation processes, lead to the occurrence of instabilities of initially uniform dislocation distributions. These instabilities result to the development o...
Article
We present here a novel multiscale modelling approach to investigate the conditions for atomic cluster self-organization on atomically flat substrates during epitaxial deposition processes. A phase field model is developed for the free energy of the system, which includes short-range as well as long-range elastic interactions between deposited atom...
Article
The evolution of a monoatomic layer, deposited on a substrate, is described by a dynamical model of the reaction-diffusion type. This model takes into account the possible coexistence of two types of gains with different orientations with respect to the substrate. It combines reaction (adsorption and desorption) and nonlinear diffusion terms, which...
Article
It is shown that coverage evolution, during atomic deposition on a substrate, may be described, on mesoscopic scales, by dynamical models of the reaction–diffusion type. The models combine reaction terms representing adsorption–desorption processes and nonlinear diffusion terms of the Cahn–Hilliard type. The combination may lead, below a critical t...
Article
Spatio-temporal pattern formation in physico-chemical systems far from thermal equilibrium has long been a puzzling phenomenon. Until the last decade, understanding pattern selection and stability mechanisms was considered as a challenge. Fortunately, thanks to intensive theoretical and experimental research, a unified framework is now available to...
Chapter
Coverage evolution, during atomic deposition on a substrate may be described, on mesoscopic scales, by dynamical models of the reaction-diffusion type, which combine reaction terms representing chemical processes such as adsorption-desorption and nonlinear diffusion terms. Below a critical temperature, uniform deposited layers are unstable, which l...
Article
The deposition of monoatomic layers on a substrate is described by a reactiondiffusion model which combines atomic adsorption, desorption and nonlinear diffusion. Close to the adsorbed layer instability, this dynamics is of the Cahn-Hilliard type, supplemented by linear reaction terms representing adsorption-desorption processes. The instability of...
Article
It is shown that coverage evolution during atomic deposition on a substrate may be described, on mesoscopic scales, by dynamic models of the reaction-diffusion type. Such models combine reaction terms representing adsorption-desorption and chemical processes and nonlinear diffusion terms which are of the Cahn-Hilliard type. This combination may lea...
Article
We report theoretical and numerical results on convection for a magnetic fluid in a viscoelastic carrier liquid. We focus on the stationary convection for idealized boundary conditions. We obtain explicit expressions of convective thresholds in terms of the control parameters of the system. Close to bifurcation, the coefficients of the correspondin...
Article
It is shown how that the combination of atomic deposition and nonlinear diffusion may lead, below a critical temperature, to the growth of nonuniform layers on a substrate. The dynamics of such a system is of the Cahn–Hilliard type, supplemented by reaction terms representing adsorption–desorption processes. The instability of growing uniform layer...
Article
We analyze the formation and selection of self-organized defect microstructure in irradiated materials within the framework of a kinetic model for point and clustered defects. We take explicitly into account the influence of glissile interstitial clusters on the stability and morphology of ordered microstructure. Under void growth conditions, we fi...
Article
Full-text available
Following the insight of P. Kolodner (J. Non-Newtonian Fluid Mech. 75 (1998) 167)on DNA solutions, a study of the Rayleigh-Bénard instability for a binary viscoelastic fluid is presented. We emphasize the role of the relaxation time typical of the Oldroyd-B model for a viscoelastic fluid and the Soret effect characteristic of binary fluids. Critica...
Article
Full-text available
We study the role of a direct intracavity polarization coupling in the dynamics of transverse pattern formation in type-II optical parametric oscillators. Transverse intensity patterns are predicted from a stability analysis, numerically observed, and described in terms of amplitude equations. Standing wave intensity patterns for the two polarizati...
Article
It is shown how the combination of atomic deposition and nonlinear diffusion may lead, below a critical temperature, to the growth of nonuniform layers on a substrate. The dynamics of such a system is of the Cahn–Hilliard type, supplemented by reaction terms representing adsorption–desorption processes. The instability of uniform layers leads to th...
Article
We study pattern formation associated with the polarization degree of freedom of the electric field amplitude in a mean field model describing a nonlinear Kerr medium close to a two-photon resonance, placed inside a ring cavity with flat mirrors and driven by a coherent x-polarized plane-wave field. In the self-focusing case, for negative detunings...
Article
Modeling spatio-temporal pattern formation in complex systems far from thermal equilibrium has long been considered as a challenge. Fortunately, during the last decade, a unified framework, which allows the study of generic aspects of pattern formation phenomena, emerged from intensive theoretical and experimental research. It has been successfully...
Article
The effect of crystal structure on laser induced deformation patterns in thin films and surfaces is analyzed within the framework of a dynamical model for the coupled evolution of defect densities and deformation fields. In crystals with covalent bonding, such as Si and SiC, preferential bond breaking may occur, as a result of the relative orientat...
Article
Strain localization and dislocation microstructure formation are typical features of plastic deformation in metals and alloys. Plastic deformation occurs by the glide of dislocations, and, although dislocation distributions are rather uniform at its onset, they usually become unstable when deformation proceeds and undergo successive transitions tow...
Article
The effect of a temporal modulation at three times the critical frequency on a Hopf bifurcation is studied in the framework of amplitude equations. We consider a complex Ginzburg-Landau equation with an extra quadratic term, resulting from the strong coupling between the external field and the unstable modes. We show that, by increasing the intensi...
Article
Full-text available
Irradiation of materials by energetic particles (e.g., electrons, ions and neutrons) is associated with very high internal power dissipation, which can drive the underlying nano- and microstructure far from normal equilibrium conditions. One of the most unusual responses in this connection is the ability of the material's nano- and microstructure t...
Article
The threshold for oscillatory convection in Rayleigh–Bénard experiments with viscoelastic binary fluids is explicitly determined as a function of separation ratio and rheological parameters. In particular, it is shown that the critical oscillation frequency may differ by several orders of magnitude on varying separation ratio and Deborah number. Th...
Article
In this paper, we analyze the onset of phase-dominant dynamics in a uniformly forced system. The study is based on the numerical integration of the Swift–Hohenberg equation and adresses the characterization of phase disorder detected from gradient computational operators as complex entropic form (CEF). The transition from amplitude to phase dynamic...
Article
The onset of convection in a binary-viscoelastic Oldroyd-B fluid is investigated. The threshold for oscillatory convection is calculated. It is shown that the critical oscillation frequency may differ by several orders of magnitude on varying separation ratio and Deborah number. The results suggest that binary fluid aspects may not be discarded whe...
Chapter
Pattern selection and stability in polymeric fluid convection are studied in the framework of amplitude equations derived in the vicinity of oscillatory instability. These are complex Ginzburg-Landau equations, and their coefficients, as computed from the underlying Navier-Stokes equations, are such that the selected patterns correspond to standing...
Article
Full-text available
We study the nature of the instability of the homogeneous steady states of the subcritical real Ginzburg-Landau equation in the presence of group velocity. The shift of the absolute instability threshold of the trivial steady state, induced by the destabilizing cubic nonlinearities, is confirmed by the numerical analysis of the evolution of its per...
Article
Convection in viscoelastic fluids may be induced by oscillatory or stationary instabilities. When the oscillatory instability appears first, the system may be described, in its vicinity by coupled Ginzburg–Landau equations. We study the convective and absolute instabilities of the steady states of these equations. We analyze pattern selection and s...
Article
The coupling between surface deformation and defect motion may be at the origin of deformation patterns in thin films under laser irradiation. We analyze the dynamics of laser-induced vacancy densities and deformation fields and show how it triggers deformational instabilities, in the case of uniform and focused laser irradiation. Pattern selection...
Article
Pattern formation in nonlinear optical cavities, when an advection-like term is present, is analyzed. This term breaks the space inversion symmetry causing the existence of a regime of convective instabilities, where noise-sustained structures can be found, and changing the pattern orientation and the selected wavevector. The concepts of convective...
Article
The formation of laser-induced deformation patterns on thin films and surfaces may be described by a dynamical model for the coupled evolution of defect densities and deformation fields of the material. Increasing laser intensity induces deformational instability, which may be characterized in the framework of linear stability analysis of undeforme...
Article
The formation and evolution of defect microstructures in irradiated materials is analysed in the framework of a dynamical model for the evolution of the two fundamental defects of irradiated microstructures, namely vacancy and interstitial clusters. The effects of irradiation on materials is described by dynamical equations for two mobile atomic si...
Article
The problem of two-dimensional (2D), transverse, noise-sustained pattern formation is theoretically and numerically studied, in the case of an optical parametric oscillator, for negative signal detuning. This gives a complete analysis of a 2D, convective, pattern forming system which is also relevant to more general 2D physical systems. For the opt...
Article
The effect of walk-off in pattern selection in optical parametric oscillators is theoretically examined. We show that a dynamic mechanism also allows us to observe the formation of structures for positive signal detunings. In this regime the pattern that is generated is a periodic array of kinks that separate regions in which one of two stable stea...
Article
Full-text available
The existence of macroscopic noise-sustained structures in nonlinear optics is theoretically predicted and numerically observed, in the regime of convective instability. The advection-like term, necessary to turn the instability to convective for the parameter region where advection overwhelms the growth, can stem from pump beam tilting or birefrin...
Article
Full-text available
We study spatiotemporal pattern formation associated with the polarization degree of freedom of the electric field amplitude in a mean field model describing a Kerr medium in a cavity with flat mirrors and driven by a coherent plane-wave field. We consider linearly as well as elliptically polarized driving field and situations of self-focusing and...
Article
The mechanical behavior of thin films subjected to laser irradiation is described by a dynamical model that is based on coupled evolution equations for the deformation and vacancy density fields. Lattice vacancies are generated in a thin layer as a result of shallow absorption of electromagnetic laser radiation. The strain field associated with lat...
Article
Evidence of noise-sustained patterns in nonlinear optical systems is given. They are found in passive optical cavities, filled by Kerr type nonlinear media, when the angle of incidence of the pump beam is not zero, in a regime of convective instability. These patterns arise as a macroscopic manifestation of dynamically amplified noise, with amplifi...
Article
The coupling between surface deformation and defect dynamics may be at the origin of deformation patterns in thin films under laser irradiation. We analyze a simple model describing the dynamics of such systems in the case of focused laser irradiation. We show, through linear, nonlinear, and numerical analysis, how rose deformation patterns, with t...
Article
Recent computer simulations on dislocation patterning have provided remarkable results in accordance with empirical laws. Moreover, several analytical models on dislocation dynamics have provided qualitative insight on dislocation patterning. However, a model, based on partial differential equations, which gives a dynamical evolution of dislocation...
Article
The effect of mean flows on pattern stability in systems where oscillatory instabilities of the Hopf type interact with stationary ones is investigated. In particular, it is shown that pattern selection may be strongly modified when the absolute instability threshold of the trivial uniform steady state is rejected beyond the stationary instability....
Chapter
One of the most intriguing aspects of the complex dynamics that govern natural phenomena is perhaps the occurrence of instabilities and symmetry breakings leading to the formation of coherent spatio-temporal structures on macroscopic scales. It is clear now that when a physico-chemical system is maintained in far-fromthermal equilibrium by the appl...
Chapter
In Chapter 1 we saw how the behavior of complex systems can be described by simple amplitude equations near primary spatial and temporal instabilities. These equations are found either by the adiabatic elimination of stable modes or by projecting the original dynamical system on its center manifold and removing non-secular terms in the resulting eq...
Chapter
As emphasized earlier in this book, stationary spatial structures may be induced by the interplay between reaction and diffusion in nonlinear chemical systems, and this pattern formation mechanism was proposed by A. Turing as early as 1952 [6.1]. Since then, various kinetic models have been dedicated to the justification of this hypothesis and to t...
Chapter
As discussed in previous chapters, the spontaneous nucleation of spatio-temporal patterns in systems driven far from thermal equilibrium by external constraints is the subject of intensive experimental and theoretical research. These studies, which were at first considered related to fundamental problems, appear now to be also of technological impo...
Chapter
As mentioned earlier, it is now well documented that many particle-irradiated materials present several types of microstructures, which correspond to the spatial organization of defect populations. The best known examples are void [11.1–11.3] and bubble lattices [11.4–11.6], precipitate ordering [11.7], defect walls, and vacancy loop ordering [11.1...
Chapter
The Rayleigh-Bénard instability, which is perhaps the most popular one in the field of pattern formation, was discovered at the beginning of the twentieth century and triggered an enormous amount of work. It played an essential role in the development of new theoretical and experimental methods in this field, triggered by the possibility of precise...
Article
We investigate a pattern-forming system close to a Hopf bifurcation with broken translational symmetry. In one-dimensional geometries, its evolution is governed by two coupled complex Ginzburg-Landau equations which describe the amplitude of the counterpropagating traveling waves that develop beyond the instability. The convective and absolute inst...
Chapter
The various phenomena discussed in the preceding chapters are among the large number of problems described by systems of nonlinear differential equations which depend on one or more control parameters (e.g., the temperature gradient across the boundaries of a Rayleigh-Bénard cell, the angular velocity of the cylinders of a Taylor-Couette apparatus,...
Chapter
It is now well documented that, despite their complexity, the dynamics of physicochemical systems driven away from thermal equilibrium may be reduced, close to bifurcation points, to much simpler forms describing the universal properties of the spatio-temporal patterns. One of the most celebrated instabilities in this framework corresponds to the H...
Chapter
Besides generic properties that only depend on the symmetries of the system, the type of bifurcation encountered, the structure of the dynamics, and so forth, non-equilibrium patterns may however also present specific properties induced by particular dependences of the kinetic coefficients on the control parameters or driving fields, or geometrical...
Chapter
In experimental systems, when a spatially extended state becomes unstable by the increase of some control parameter, the new stable state may grow in different ways. It may grow globally from the noise that is always present in macroscopic systems, or it may propagate from special spatial locations such as impurities, nonuniformities, boundaries, a...
Chapter
Spatially or temporally modulated systems are very common in nature, and the effect of external fields on these systems has long been studied. For example, the forcing of a large variety of nonlinear oscillators, from the pendulum to Van der Pol or Duffing oscillators, has led to detailed studies of the different temporal behaviors that were obtain...
Chapter
Strain localization and dislocation microstructure formation are typical features of the plastic deformation of metals and alloys [12.1]. Plastic deformation occurs by the glide of dislocations, and, although the dislocation distributions are rather uniform at its onset, they usually become unstable when deformation proceeds and undergo successive...
Article
Pattern selection and stability in viscoelastic convection are studied in the framework of amplitude equations derived in the vicinity of stationary and oscillatory instabilities. The oscillatory instability corresponds to a Hopf bifurcation with broken translational symmetry. When this instability is the first to appear with increasing Rayleigh nu...
Article
A general theory for the spatial ordering of immobile clustered defects in irradiated materials is presented here. A vectorial form for the Fourier transforms of perturbations in the concentration of point and clustered defects is derived. Linear stability analysis indicates that, under conditions appropriate for void growth (high temperature), ins...
Article
We study the effect of spatial frequency-forcing on standing-wave solutions of coupled complex Ginzburg-Landau equations. The model considered describes several situations of nonlinear counterpropagating waves and also of the dynamics of polarized light waves. We show that forcing introduces spatial modulations on standing waves which remain freque...
Chapter
Physico-chemical systems driven away from thermal equilibrium usually undergo various types of instabilities leading to the formation of spatio-temporal patterns on macroscopic time and space scales. In two and three-dimensional geometries, patterns of different symmetries may be simultaneously stable. The phase stability of these patterns and diff...
Article
We analyze the formation and evolution of point and line defect microstructure in irradiated materials. The effects of irradiation on materials are described by dynamical conservation equations for two mobile atomic size species (vacancies and interstitial atoms), and two basic immobile elements of the microstructure (vacancy and interstitial clust...
Chapter
Full-text available
Pattern forming instabilities in spatially extended dissipative systems driven away from equilibrium have been the focus of a large activity for many years. The goal of this chapter is to present some theoretical concepts that have been developed to understand and describe these dissipative structures [1] from a macroscopic point of view. Although...
Article
Close to a supercritical Hopf bifurcation, oscillatory media may be described, by the complex Ginzburg-Landau equation. The most important spatiotemporal behaviors associated with this dynamics are reviewed here. It is shown, on a few concrete examples, how real chemical oscillators may be described by this equation, and how its coefficients may be...
Article
This volume is devoted to the presentation of the main contributions to the workshop "From oscillations to excitability: A case study in spatially extended systems," organized by the authors in Nice in June 1993. It gives an overview of the current research on spatiotemporal patterns in a wide range of systems that display self-oscillatory or excit...
Chapter
The study of the different patterns which appear beyond the electrohydrodynamic instability of nematic liquid crystals is performed in the framework of a dynamical model of the Proctor-Sivashinsky type. This model describes the experimentally observed transitions between rolls, zig-zag and bimodal structures.
Article
The application of external fields on spatio-temporal Hopf bifurcations is known to modify the stability of the associated wave patterns. In particular, left and right travelling waves are linearly coupled by pure temporal modulations provided the frequency of these modulations is close to twice the critical ones, and travelling waves may be transf...
Article
We study analytically and numerically the tridimensional pattern selection problem for reaction-diffusion systems. Qualitative agreement is found with the recent experimental results.
Article
The effect of temporal modulations on wave patterns induced by spatiotemporal Hopf bifurcations is discussed in the framework of amplitude equations of the Ginzburg-Landau type. The approach is well adapted to the study of pattern formation in liquid crystals which, on the other hand, provide a class of easily forced systems. A few examples, relate...
Article
Full-text available
In the mean field description of the incommensurate transition of quartz, the triangular 3-$q$, the striped 1-$q$ and the $\beta$ phases meet at the same critical point. It is shown that critical fluctuations lift this degeneracy and transform the second order transition into first order ones. Furthermore, there is a regime of the parameter space w...
Article
The effect of external fields on non-equilibrium structures is discussed in the framework of amplitude equations of the Ginzburg-Landau type. This approach is well adapted to the study of pattern formation in liquid crystals which, on the other hand, provide a class of easily forced systems. A few examples, related to experimental observations, are...
Chapter
Defects are known to play an important role in the macroscopic behavior of equilibrium and non equilibrium structures. They are able to disorganize spatio-temporal patterns, to trigger transitions between patterns of different symmetries, but also to form various kinds of microstructures. In particular, the occurence of spatial instabilities leadin...
Chapter
Beyond various pattern forming instabilities structures with different symmetries may be simultaneously stable. Several aspects of the transitions between such structures are studied in the framework of amplitude equations of the Ginzburg-Landau type. In particular, it is shown how boundary effects, external fields and internal fluctuations may aff...
Article
Dislocation patterns arise in a large variety of situations where crystalline materials are submitted to mechanical constraints such as cyclic or monotonous loadings. It may be shown that, under the combined effect of defect motion and interactions, uniform dislocation distribution may become unstable versus spatial modulations leading to the forma...
Chapter
During the transformation of the vein structure into the persistent slip hand structure in a copper crystal under fatigue (i.e. cyclic stress) the dislocation distribution undergoes complex microscopic changes. A roughly homogeneous distribution is converted into a periodic roll structure of many hundreds of wavelengths that crosses the crystal thr...
Chapter
The development of a dislocation inhomogeneity in the form of regular spatial patterns during cyclic deformation is examined from a dynamical instability point of view. A gradient-dependent dislocation dynamics framework is devised and the competition between mobility, interaction, and generation processes is considered. Within such a framework, it...
Chapter
One of the most natural and still intriguing behavior of complex physico-chemical systems driven sufficiently far from thermal equilibrium is their ability to undergo symmetry-breaking instabilities leading to the spontaneous formation of coherent structures over macroscopic time and space scales (Nicolis and Prigogine, 1977). Such a behavior has b...
Chapter
Spatial instabilities leading to the formation of defect patterns and microstructures seem to appear generically in solids driven away from thermal equilibrium by physicochemical constraints. Some of these instabilities have been recently studied within the framework of dynamical models for the defect densities. The basic properties of these models...
Chapter
The origin of defect patterns and microstructures in driven materials has recently been studied within the framework of dynamical models for the defect densities. These models take into account the motion and interaction of defects and may present various types of pattern forming instabilities. Some basic properties of these models are reviewed whe...
Book
Understanding the origin of spatio-temporal order in open systems far from thermal equilibrium and the selection mechanisms of spatial struc­ tures and their symmetries is a major theme of present day research into the structures of continuous matter. The development of methods for pro­ ducing spatially ordered microstructures in solids by non-equi...

## Citations

... Moreover, self-organized nano-structures may emerge under some specific conditions of irradiation flux and temperature. Many examples of such patterns have been observed in metals, alloys, ceramics and amorphous materials [3][4][5][6][7]. ...
... Self-organization is one method to produce characteristic structures and several selforganization phenomena of defect clusters under high-energy particle irradiations such as Self-organization is one method to produce characteristic structures and several selforganization phenomena of defect clusters under high-energy particle irradiations such as voids, bubbles, and stacking fault tetrahedra have been reported so far [12][13][14][15][16]. Spontaneous well-ordered periodicity can be developed on a broad surface by ion beam sputtering and a numerical model has been proposed as the formation process [17]. ...
... In this context, excitable waves can emerge in extended systems which are locally excitable [7,25,26]. However, the spatial coupling can be responsible for the coherent structures emerging from the spiking dynamics even in systems that are non locally excitable [27,28], and for excitable-like behaviors which stem from front interactions [29]. Recent works have also focused on the characterization of travelling pulses in type-I excitable media [30,31]. ...