# Orazio DescalziUniversity of the Andes (Chile) | UANDES · Faculty of Engineering and Applied Sciences

Orazio Descalzi

Dr. rer. nat. Univ. Essen, Germany

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87

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Introduction

**Skills and Expertise**

## Publications

Publications (87)

The purpose of this article is twofold. Firstly, to investigate the formation of localized spatiotemporal chaos in the complex cubic Ginzburg–Landau equation including nonlinear gradient terms. We found a transition to spatiotemporal disorder via quasiperiodicity accompanied by the fact that incommensurate satellite peaks arise around the fundament...

We investigate properties of oscillatory dissipative solitons (DSs) in a cubic complex Ginzburg-Landau equation stabilized by nonlinear gradient terms. As a main result we find a transition to dissipative solitons with spatiotemporal disorder as a function of the diffusion coefficient. This transition proceeds via quasiperiodicity and shows incomme...

We investigate the influence of spatially homogeneous multiplicative noise on propagating dissipative solitons (DSs) of the cubic complex Ginzburg-Landau equation stabilized by nonlinear gradient terms. Here we focus on the nonlinear gradient terms, in particular on the influence of the Raman term and the delayed nonlinear gain. We show that a fair...

We study the interaction of stable dissipative solitons of the cubic complex Ginzburg-Landau equation which are stabilized only by nonlinear gradient terms. In this paper we focus for the interactions in particular on the influence of the nonlinear gradient term associated with the Raman effect. Depending on its magnitude, we find up to seven possi...

The propagation of light pulses in dual-core nonlinear optical fibers is studied using a model proposed by Sakaguchi and Malomed. The system consists of a supercritical complex Ginzburg–Landau equation coupled to a linear equation. Our analysis includes single standing and walking solitons as well as walking trains of 3, 5, 6, and 12 solitons. For...

We show that for a large range of approach velocities and over a large interval of stabilizing cubic cross-coupling between counterpropagating waves, a collision of stationary pulses leads to a partial annihilation of pulses via a spontaneous breaking of symmetry. This result arises for coupled cubic-quintic complex Ginzburg-Landau equations for tr...

We study a single cubic complex Ginzburg–Landau equation with nonlinear gradient terms analytically and numerically. This single equation allows for the existence of stable dissipative solitons exclusively due to nonlinear gradient terms. We shed new light on the feedback loop, leading to dissipative solitons (DSs) by analyzing a mechanical analog...

We present a feedback mechanism for dissipative solitons in the cubic complex Ginzburg-Landau (CGL) equation with a nonlinear gradient term. We are making use of a mechanical analog containing contributions from a potential and from a nonlinear viscous term. The feedback mechanism relies on the continuous supply of energy as well as on dissipation...

We investigate the simultaneous influence of spatially homogeneous multiplicative noise as well as of spatially δ-correlated additive noise on the formation of localized patterns in the framework of the cubic-quintic complex Ginzburg-Landau equation. Depending on the ratio between the strength of additive and multiplicative noise we find a number o...

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...

The influence of additive noise, multiplicative noise, and higher-order effects on exploding solitons in the framework of the prototype complex cubic-quintic Ginzburg-Landau equation is studied. Transitions from explosions to filling-in to the noisy spatially homogeneous finite amplitude solution, collapse (zero solution), and periodic exploding di...

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...

We give an overview of the influence of noise on spatially localized patterns and their interaction. Localized patterns include stationary dissipative solitons, oscillatory dissipative solitons with one and two frequencies as well as exploding 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 char...

We demonstrate the occurrence of anomalous diffusion of dissipative solitons in a “simple” and deterministic prototype model: the cubic-quintic complex Ginzburg-Landau equation in two spatial dimensions. The main features of their dynamics, induced by symmetric-asymmetric explosions, can be modeled by a subdiffusive continuous-time random walk, whi...

We show the existence of periodic exploding dissipative solitons. These nonchaotic explosions appear when higher-order nonlinear and dispersive effects are added to the complex cubic-quintic Ginzburg-Landau equation modeling fiber soliton lasers. This counterintuitive phenomenon is the result of period-halving bifurcations leading to order (periodi...

We show the existence of periodic exploding dissipative solitons. These non-chaotic explosions appear when higher-order nonlinear and dispersive effects are added to the complex cubic quintic Ginzburg Landau equation modelling soliton transmission lines. This counterintuitive phenomenon is the result of period-halving bifurcations leading to order...

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 counterpropagating DSs, we find non-uni...

We investigate the collisions of two counter-propagating exploding dissipative solitons (DSs). We demonstrate that six different outcomes can occur as a function of the nonlinear cross-coupling between the counter-propagating waves: complete interpenetration, one compound exploding DS as well as four types of two compound DSs that can be stationary...

This article shows for the first time the existence of periodic exploding dissipative solitons. These non-chaotic explosions appear when higher-order non-linear and dispersive effects are added to the complex cubic–quintic Ginzburg–Landau equation modeling soliton transmission lines. This counter-intuitive phenomenon is the result of period-halving...

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 regi...

We review the work on exploding dissipative solitons in one and two spatial dimensions. Features covered include: the transition from modulated to exploding dissipative solitons, the analogue of the Ruelle-Takens scenario for dissipative solitons, inducing exploding dissipative solitons by noise, two classes of exploding dissipative solitons in two...

We study the interaction of quasi-one-dimensional (quasi-1D) dissipative solitons (DSs). Starting with quasi-1D solutions of the cubic-quintic complex Ginzburg-Landau (CGL) equation in their temporally asymptotic state as the initial condition, we find, as a function of the approach velocity and the real part of the cubic interaction of the two cou...

We study the influence of an analog of self–steepening (SST), which is a term breaking the T →−T symmetry, on explosive localized solutions for the cubic–quintic complex Ginzburg–Landau equation in the anomalous dispersion regime. We find that while this explosive behavior occurs for a wide range of the parameter s, characterizing SST, the mean dis...

Noise is an everywhere phenomenon. Its influence could be described theoretically quite easily, but is hard to measure in an experiment. Catalytic reactions on surfaces can be described by nonlinear reaction-diffusion equations. For one of such surface reactions -CO oxidation on Iridium(111) surfaces -the probability distribution of CO 2 rates arou...

We show that exploding dissipative solitons can arise in a reaction-diffusion system for a range of parameters. As a function of a vorticity parameter, we observe a sequence of transitions from oscillatory localized states via meandering dissipative solitons to exploding dissipative solitons propagating in one direction for long times followed by t...

Dissipative solitons show a variety of behaviors not exhibited by their conservative counterparts. For instance, a dissipative soliton can remain localized for a long period of time without major profile changes, then grow and become broader for a short time-explode-and return to the original spatial profile afterward. Here we consider the dynamics...

We describe the stable existence of quasi-one-dimensional solutions of the two-dimensional cubic-quintic complex Ginzburg-Landau equation for a large range of the bifurcation parameter. By quasi-one-dimensional (quasi-1D) in the present context, we mean solutions of fixed shape in one spatial dimension that are simultaneously fully extended and spa...

We investigate a two-dimensional extended system showing chaotic and localized structures. We demonstrate the robust and stable existence of two types of exploding dissipative solitons. We show that the center of mass of asymmetric dissipative solitons undergoes a random walk despite the deterministic character of the underlying model. Since dissip...

In this article we consider the CO oxidation on Ir(111) surfaces under large external noise with large autocorrelation imposed on the composition of the feed gas, both in experiments and in theory. We report new experimental results that show how the fluctuations force the reaction rate to jump between two well defined states. The statistics of the...

We investigate the transition to explosive dissipative solitons and the destruction of invariant tori in the complex cubic-quintic Ginzburg-Landau equation, in the regime of anomalous linear dispersion, as a function of the distance from linear onset. Using Poncar´ e sections we sequentially find fixed points, quasiperiodicity (two incommesurate fr...

We study the influence of noise on the spatially localized, temporally regular states (stationary, one frequency, two frequencies) in the regime of anomalous dispersion for the cubic-quintic complex Ginzburg-Landau equation as a function of the bifurcation parameter. We find that noise of a fairly small strength η is sufficient to reach a chaotic s...

We study the stationary solutions of the real Ginzburg–Landau equation with periodic boundary conditions in a finite box. We show explicitly how to construct nucleation solutions allowing transitions between stable plane waves.

It is shown that pulses in the complete quintic one-dimensional Ginzburg–Landau equation with complex coefficients appear through a saddle-node bifurcation which is determined analytically through a suitable approximation of the explicit form of the pulses. The results are in excellent agreement with direct numerical simulations.

We investigate the route to exploding dissipative solitons in the complex cubic-quintic Ginzburg-Landau equation, as the bifurcation parameter, the distance from linear onset, is increased. We find for a large class of initial conditions the sequence: stationary localized solutions, oscillatory localized solutions with one frequency, oscillatory lo...

Systems driven far from thermodynamic equilibrium can create dissipative structures through the spontaneous breaking of symmetries. A particularly fascinating feature of these pattern-forming systems is their tendency to produce spatially confined states. These localized wave packets can exist as propagating entities through space and/or time. Vari...

We investigate the properties of and the transition to exploding dissipative solitons as they have been found by Akhmediev's group for the cubic-quintic complex Ginzburg-Landau equation. Keeping all parameters fixed except for the distance from linear onset, μ , we covered a large range of values of μ from very negative values to μ=0 , where the ze...

We study the effect of external noise on the catalytic oxidation of CO on an Iridium(111) single crystal under ultrahigh vacuum conditions. This reaction can be considered as a model of catalysis used in the industry. In the absence of noise, the reaction exhibits one or two stable stationary states, depending on the control parameters such as temp...

We investigate the influence of Dirichlet boundary conditions on various types of localized solutions of the cubic-quintic complex Ginzburg-Landau equation as it arises as an envelope equation near the weakly inverted onset of traveling waves. We find that various types of nonmoving pulses and holes can accommodate Dirichlet boundary conditions by...

We study the effect of external noise on the catalytic oxidation of CO on an Iridium( 111) single crystal under ultrahigh vacuum conditions. This reaction can be considered as a model of catalysis used in industry. In the absence of noise the reaction exhibits one or two stable stationary states, depending on control parameters such as temperature...

The cubic-quintic complex Ginzburg-Landau is the amplitude equation for systems in the vicinity of an oscillatory sub-critical bifurcation (Andronov-Hopf), and it shows different localized structures. For pulse-type localized structures, we review an approximation scheme that enables us to compute some properties of the structures, like their exist...

We show and characterize numerically moving breathing pulses in the one-dimensional complex cubic-quintic Ginzburg-Landau equation. This class of stable moving breathing pulses has not been described before for this prototype envelope equation as it arises near the weakly hysteretic onset of traveling waves.

We study the process of localization of a hexagonal pattern in a uniform background, specifically, the role played by the shape and size of the domain where the hexagonal pattern is confined. We base our analysis on a numerical study of a Swift-Hohenberg type equation (which exhibits coexistence between hexagons and a uniform state), and in a scale...

Partial annihilation of two counterpropagating dissipative solitons, with only one pulse surviving the collision, has been widely observed in different experimental contexts, over a large range of parameters, from hydrodynamics to chemical reactions. However, a generic picture accounting for partial annihilation is missing. Based on our results for...

A classical geometry, widely observed in systems far from equilibrium, is the formation of hexagonal patterns. Using a prototype
Swift-Hohenberg equation for the order parameter we study the localization mechanism for hexagons surrounded by a uniform
phase. Numerical simulations show that the existence range for localized structures depends on the...

We investigate the influence of the boundary conditions and the box size on the existence and stability of various types of
localized solutions (particles and holes) of the cubic-quintic complex Ginzburg-Landau equation as it arises as a prototype
envelope equation near the weakly hysteretic onset of traveling waves. Two types of boundary condition...

The dynamics of perturbations around sinks and sources of traveling waves (TW) is studied in the cubic-quintic Ginzburg–Landau equation from an analytical point of view. Perturbations generically propagate in a direction opposite to the TW. Thus, a sink of TW is a source of perturbations and vice versa. For small values of time we predict there is...

We discuss the results of the interaction of counter-propagating pulses for two coupled complex cubic-quintic Ginzburg-Landau equations as they arise near the onset of a weakly inverted Hopf bifurcation. As a result of the interaction of the pulses we find in 1D for periodic boundary conditions (corresponding to an annular geometry) many different...

In this contribution we compare the properties of hole solutions of the cubic complex Ginzburg‐Landau equation with those of the cubic‐quintic complex Ginzburg‐Landau (CGL) equation in one spatial dimension. Both equations occur as prototype envelope equations near the onset of an oscillatory bifurcation to traveling waves.
While hole solutions of...

Given a system in the vicinity of an oscillatory subcritical bifurcation (Hopf) that presents localized structures, we model its dynamics with the Quintic Complex Ginzburg-Landau Equation. For pulse-type structures, we study the bifurcation to fronts via a quasi-analytical approach.

We study the interaction of counterpropagating pulse solutions for two coupled complex cubic-quintic Ginzburg-Landau equations in an annular geometry. For small approach velocity we find as an outcome of such collisions several results including zigzag bound pulses, stationary bound states of 2pi holes, zigzag 2pi holes, stationary bound states of...

We investigate in the framework of the quintic complex Ginzburg–Landau (CGL) equation in one spatial dimension the dynamics of the transition from moving pulse solutions to moving hole solutions, a new class of solutions found for this equation very recently. We find that the transition between these two classes of solutions is weakly hysteretic an...

We study numerically a prototype equation which arises generically as an envelope equa- tion for a weakly inverted bifurcation associated to traveling waves: The complex quintic Ginzburg{Landau equation. We show six di erent stable localized structures including stationary pulses, moving pulses, stationary holes and moving holes, starting from lo-...

We show numerically different stable localized structures including stationary holes, moving holes, breathing holes, stationary and moving pulses in the one-dimensional subcritical complex Ginzburg-Landau equation with periodic boundary conditions, and using two classes of initial conditions. The coexistence between different types of stable soluti...

By means of a matching approach we study analytically the appearance of static and oscillating-modulus pulses in the one-dimensional quintic complex Ginzburg-Landau equation without nonlinear gradient terms. When considering nonlinear gradient terms the method enables us to calculate the velocities of the stable and unstable moving pulses. We focus...

We present a simple reaction-diffusion model for two variables. The model was originally designed to have a stable localized solution for a range of parameters as a consequence of the coexistence of a stable limit cycle and a stable fixed point. We classify the spatially homogeneous solutions of the model. In addition we describe several bifurcatio...

We show numerically that the one-dimensional quintic complex Ginzburg-Landau equation admits four different types of stable hole solutions. We present a simple analytic method which permits to calculate the region of existence and approximate shape of stable hole solutions in this equation. The analytic results are in good agreement with numerical...

Oscillating localized structures are studied for a simple reaction–diffusion model from an analytical point of view. The result is a particle solution which acts as a source of traveling waves. The analytical expressions obtained are in good agreement with direct numerical simulations.

We study analytically a system sustaining stable moving localized structures, namely, the one-dimensional quintic complex Ginzburg–Landau (G–L) equation with non-linear gradients. We obtain approximate solutions for the stable moving pulse and its velocity. The results are in excellent agreement with direct numerical simulations.

We present a simple autocatalytic reaction-diffusion model for two variables, which shows for fixed parameter values the simultaneous stable coexistence of particle solutions as well of two types of hole solutions. The associated spatially homogeneous system is characterized by the coexistence of one stable fixed point and a stable limit cycle solu...

A simple reaction-diffusion model, which admits stable oscillating localized structures is discussed. The approximate analytical expressions for localized oscillating structures in the reaction-diffusion model are calculated using a generalized matching approach. It is found that the oscillating particlelike solutions lead to the traveling waves ge...

We study the one-dimensional supercritical real and complex subcritical Ginzburg- Landau equations. In the real case, with
periodic boundary conditions in a finite box, we construct analytically nucleation solutions allowing transitions between
stable plane waves. Moreover we construct vortex solutions and study the Eckhaus bifurcation in a finite...

Part I: Review Articles. Transitions between Spatio- Temporal Patterns in Non Equilibrium Systems D. Walgraef. Galilean and Relativistic Nonlinear Wave Equations: An Hydrodynamic Tool? M. Abid, et al. Part II: Instabilities and Pattern Formation. Chaotically Induced Defect Diffusion P. Coullet, K. Emilsson. Pattern Formation and Phase Turbulence in...

We study analytically the asymptotic linear stability of fixed-modulus dissipative–dispersive localized solutions of the one-dimensional quintic complex Ginzburg–Landau (GL) equation in the region where there exists a coexistence of homogeneous attractors. The linear analysis gives an indication for the existence of pulses with an oscillating modul...

We study stationary, localized solutions in the complex subcritical Ginzburg-Landau equation in the region where there exists coexistence of homogeneous attractors. Using a matching approach, we report on the fact that the appearance of pulses are related to a saddle-node bifurcation. Numerical simulations are in good agreement with our theoretical...

The notion of non-equilibrium potential for systems far from equilibrium is reviewed and the relation to the reversed process is examined. The potential is constructed in the neighborhood of the homogeneous attractors for a non-variational extended system, namely the subcritical complex Ginzburg–Landau equation. This construction is the second know...

The stationary probability is analytically determined in a suitable approximation for several systems presenting a weak noise transition. Numerical simulations are presented which corroborate the analytical results.

We consider walls connecting symmetric states in nonvariational one dimensional spatially extended systems. We show that the problem can be analyzed in terms of a free energy (nonequilibrium potential), which takes the same value in the asymptotic states (x→±∞). The motion of the walls can be understood as a residual dynamics on an extended attract...

The steady state distribution functional of the supercritical complex Ginzburg-Landau equation with weak noise is determined asymptotically for long-wave-length fluctuations including the phaseturbulent regime. This is done by constructuring a non-equilibrium potential solving the Hamilton-Jacobi equation associated with the Fokker-Planck equation....

A gradient expansion is used to obtain a Lyapunov functional (the nonequilibrium potential) for the supercritical complex Ginzburg-Landau equation. The method simplifies the task of solving the Hamilton-Jacobi equation associated with the steady-state distribution of the stochastic Ginzburg-Landau equation with weak noise and it confirms and extend...

The existence of polynomial approximations for nonequilibrium potentials determined by a master equation near an instability of arbitrary codimension with diagonalizable linear part is studied. It is shown that the approximations exist, provided some relations are satisfied between the coefficients of the master equation.