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Discrete & Continuous Dynamical Systems - S

2011 , Volume 4 , Issue 5

Issue on Localized Excitations in Nonlinear Complex Systems (LENCOS'09)

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Ricardo Carretero-González , Jesús Cuevas Maraver , Dimitri J. Frantzeskakis , P.G. Kevrekidis and  Faustino Palmero Acebedo
2011, 4(5): i-iii doi: 10.3934/dcdss.2011.4.5i +[Abstract](25) +[PDF](110.0KB)
This issue of Discrete and Continuous Dynamical Systems - Series S is a compilation of papers representing the current state-of-the-art on the field of localized excitations and their role in the dynamics of complex physical systems. During the last two decades, an impressive volume of theoretical and experimental work has been devoted to the existence, stability and dynamics of such coherent structures. They have been identified as critical components of numerous continuous and discrete dynamical systems and, depending on the context (and their particular form), they may be referred to as solitons, instantons, kinks, breathers, or quodons, among many others. We nowadays think of such localized nonlinear excitations as being ubiquitous in nature due to their experimental realization in many diverse systems including, but not limited to, optical fibers and waveguide arrays, photonic crystals, Bose-Einstein condensates, molecular crystals, quasi-one-dimensional solids, Josephson-junctions and arrays thereof, layered silicates, micromechanical cantilever arrays, uranium crystals, pendulum arrays, water waves, electrical transmission lines, ferromagnetic and antiferromagnetic materials, granular crystals and so on. Additionally, they are also conjectured to play an important role in denaturation transitions and bubble formation in DNA, protein folding, atom ejection and defect migration in crystals, low-temperature reconstructive transformations, and many others. The study of nonlinear localized excitations is a long-standing challenge for research in basic and applied science, as well as engineering, due to their importance in understanding and predicting phenomena arising in nonlinear and complex systems, but also due to their potential for the development and "design" of novel applications.

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Nonlinear wave dynamics: From lasers to fluids
Mark J. Ablowitz , Terry S. Haut , Theodoros P. Horikis , Sean D. Nixon and  Yi Zhu
2011, 4(5): 923-955 doi: 10.3934/dcdss.2011.4.923 +[Abstract](27) +[PDF](1581.3KB)
Nonlinear wave equations are central to the study of nonlinear optics and fluid dynamics. Notably, recent research has shown that solitons can be generated in mode-locked lasers. An interesting application of these lasers is the development of optical clocks which have the potential to be considerably more accurate than atomic clocks. Another important area of research in nonlinear optics is lattice dynamics where localized solitary wave or solitons can be obtained in periodic and irregular lattice systems. In honeycomb lattices, discrete and continuous nonlinear Dirac systems can be derived in certain parameter regimes; the Dirac systems describe conical diffraction, a phenomena observed in recent experiments. In water and internal waves the classical equations are reformulated as a system of coupled equations where the free surface equations are formulated as nonlocal equation and the depth variable is eliminated. These systems reduce to interesting asymptotic equations in suitable limits. A numerical method, termed spectral renormalization, is used to find solitary waves in nonlinear optics, water waves and multi-fluid systems.
Analysis of supercontinuum generation under general dispersion characteristics and beyond the slowly varying envelope approximation
Alejandro B. Aceves , Rondald Chen , Yeojin Chung , Thomas Hagstrom and  Michelle Hummel
2011, 4(5): 957-973 doi: 10.3934/dcdss.2011.4.957 +[Abstract](28) +[PDF](782.8KB)
The generation of broadband supercontinua (SC) in air-silica microstructured fibers results from a delicate balance of dispersion and nonlinearity. We analyze two models aimed at better understanding SC. In the first one, we characterize linear dispersion in the Fourier domain from the calculated group velocity dispersion (GVD) without using a Taylor approximation for the propagation constant. Results of our numerical simulations are in good agreement with experiments. A novel relevant length scale, namely the length for shock formation, is introduced and its role is discussed. The second part shows similar dynamics for a model that goes beyond the slowly varying approximation for optical pulse propagation.
Asymptotics for supersonic traveling waves in the Morse lattice
Alejandro B. Aceves , Luis A. Cisneros-Ake and  Antonmaria A. Minzoni
2011, 4(5): 975-994 doi: 10.3934/dcdss.2011.4.975 +[Abstract](46) +[PDF](457.8KB)
Supersonic traveling wave solutions to the Morse lattice are considered. By numerical means, we show that initial shock like initial values always evolve into traveling shock solutions after the emission of radiation traveling at the sound speed. Using a trial function which includes a shape modulation in the core of the shock we show how the Peierls-Nabarro self consistent potential induced by the lattice is canceled by the adjustment of the phase. We find excellent agreement between the modulation analysis and the numerical solutions.
Nonsymmetric moving breather collisions in the Peyrard-Bishop DNA model
Azucena Álvarez , Francisco R. Romero , José M. Romero and  Juan F. R. Archilla
2011, 4(5): 995-1006 doi: 10.3934/dcdss.2011.4.995 +[Abstract](57) +[PDF](2406.4KB)
We study nonsymmetric collisions of moving breathers (MBs) in the Peyrard-Bishop DNA model. In this paper we have considered the following types of nonsymmetric collisions: head-on collisions of two breathers traveling with different velocities; collisions of moving breathers with a stationary trapped breather; and collisions of moving breathers traveling with the same direction. The various main observed phenomena are: one moving breather gets trapped at the collision region, and the other one is reflected; breather fusion without trapping, with the appearance of a new moving breather; and breather generation without trapping, with the appearance of new moving breathers traveling either with the same or different directions. For comparison we have included some results of a previous paper concerning to symmetric collisions, where two identical moving breathers traveling with opposite velocities collide. For symmetric collisions, the main observed phenomena are: breather generation with trapping, with the appearance of two new moving breathers with opposite velocities and a stationary breather trapped at the collision region; and breather generation without trapping, with the appearance of new moving breathers with opposite velocities. A common feature for all types of collisions is that the collision outcome depends on the internal structure of the moving breathers and the exact number of pair-bases that initially separates the stationary breathers when they are perturbed. As some nonsymmetric collisions result in the generation of a new stationary trapped breather of larger energy, the trapping phenomenon could play an important part of the complex mechanisms involved in the initiation of the DNA transcription processes.
Existence of solitary waves in nonlinear equations of Schrödinger type
Juan Belmonte-Beitia and  Vladyslav Prytula
2011, 4(5): 1007-1017 doi: 10.3934/dcdss.2011.4.1007 +[Abstract](38) +[PDF](372.0KB)
In this work we study the existence of solitary waves in nonlinear equations of Schrödinger type. We prove the existence of the positive solution and using the bifurcation theory show that the norm of the given solution tends to zero as the coefficient corresponding to the linear term vanishes.
Variational approximations of bifurcations of asymmetric solitons in cubic-quintic nonlinear Schrödinger lattices
Christopher Chong and  Dmitry Pelinovsky
2011, 4(5): 1019-1031 doi: 10.3934/dcdss.2011.4.1019 +[Abstract](40) +[PDF](382.7KB)
Using a variational approximation we study discrete solitons of a nonlinear Schrödinger lattice with a cubic-quintic nonlinearity. Using an ansatz with six parameters we are able to approximate bifurcations of asymmetric solutions connecting site-centered and bond-centered solutions and resulting in the exchange of their stability. We show that the numerical and variational approximations are quite close for solitons of small powers.
The VES hypothesis and protein misfolding
Leonor Cruzeiro
2011, 4(5): 1033-1046 doi: 10.3934/dcdss.2011.4.1033 +[Abstract](39) +[PDF](1079.3KB)
Proteins function by changing conformation. These conformational changes, which involve the concerted motion of a large number of atoms are classical events but, in many cases, the triggers are quantum mechanical events such as chemical reactions. Here the initial quantum states after the chemical reaction are assumed to be vibrational excited states, something that has been designated as the VES hypothesis. While the dynamics under classical force fields fail to explain the relatively lower structural stability of the proteins associated with misfolding diseases, the application of the VES hypothesis to two cases can provide a new explanation for this phenomenon. This explanation relies on the transfer of vibrational energy from water molecules to proteins, a process whose viability is also examined.
Sine-Gordon wobbles through Bäcklund transformations
Sara Cuenda , Niurka R. Quintero and  Angel Sánchez
2011, 4(5): 1047-1056 doi: 10.3934/dcdss.2011.4.1047 +[Abstract](38) +[PDF](377.7KB)
In this work we construct the wobble exact solution of sine-Gordon equation by means of Bäcklund Transformations. We find the parameters of the transformations corresponding to the Bianchi diagram for the wobble as a particular $3$-soliton solutions. We show that this solution agrees with the wobbles obtained by Kälbermann and Segur by means of the Inverse Scattering Transform, and by Ferreira et al. using the Hirota method. The new formulation introduced allows to identify easily the parameters that define the building blocks of this solution -- a kink and a breather, and can be used in further studies of this solution in the perturbed sine-Gordon equation.
Interaction of moving discrete breathers with interstitial defects
Jesús Cuevas , Bernardo Sánchez-Rey , J. C. Eilbeck and  Francis Michael Russell
2011, 4(5): 1057-1067 doi: 10.3934/dcdss.2011.4.1057 +[Abstract](38) +[PDF](1591.5KB)
In this paper, interstitial migration generated by scattering with a mobile breather is investigated numerically in a Frenkel-Kontorova one-dimensional lattice. Consistent with experimental results, it is shown that interstitial diffusion is more likely and faster than vacancy diffusion. Our simulations support the hypothesis that a long-range energy transport mechanism involving moving nonlinear vibrational excitations may significantly enhance the mobility of point defects in a crystal lattice.
Interaction length of DM solitons in the presence of third order dispersion with loss and amplification
Francisco J. Diaz-Otero and  Pedro Chamorro-Posada
2011, 4(5): 1069-1078 doi: 10.3934/dcdss.2011.4.1069 +[Abstract](28) +[PDF](399.5KB)
We present an analysis of the interaction properties of time-division multiplexed dispersion-managed solitons in the strong management regime. The study is based on an ordinary differential equations model, obtained by means of the variational method, which takes into account third order dispersion, loss and periodic amplification. The validity of the model is assessed by comparing the variational results with direct simulations of the underlying partial differential equations, finding excellent agreement. We first study the conditions for stable single soliton pulse propagation as the amplifier position is varied in the dispersion map. Interactions between adjacent pulses are then investigated for both lossless and lossy systems and the effect of third-order dispersion is addressed. We find that the increase found in the interaction distance can be explained by an asymmetric effective shift of the average dispersion of each of the soliton pulses induced in the interaction process.
Dynamics of edge dislocation clusters interacting with running acoustic waves
Sergey V. Dmitriev , Asiya A. Nazarova , Anatoliy I. Pshenichnyuk and  Albert M. Iskandarov
2011, 4(5): 1079-1094 doi: 10.3934/dcdss.2011.4.1079 +[Abstract](42) +[PDF](467.3KB)
Interaction of straight edge dislocation clusters with monochromatic sound wave having nonzero wavevector is investigated taking into account the dislocation mass. We report on a significant increase of drift velocities of clusters when the sound wave frequency approaches a cluster's eigenfrequency. Of practical importance is the increase of the drift velocity observed for clusters with nonzero topological charge interacting with small frequency sound waves. We also demonstrate the possibility to excite a gap discrete breather in a chain of dislocation dipoles.
The dynamics of the kink in curved large area Josephson junction
Tomasz Dobrowolski
2011, 4(5): 1095-1105 doi: 10.3934/dcdss.2011.4.1095 +[Abstract](35) +[PDF](376.2KB)
A formalism that allows description of the kink motion in an arbitrarily curved large area Josephson junction is proposed. A general formula for the lagrangian density that describes the curved Josephson junction, in small curvature regime, is obtained. Examples of propagation of the kink along the curved Josephson junction are considered.
Breathers and kinks in a simulated crystal experiment
Qingxu Dou , Jesús Cuevas , J. C. Eilbeck and  Francis Michael Russell
2011, 4(5): 1107-1118 doi: 10.3934/dcdss.2011.4.1107 +[Abstract](42) +[PDF](2611.7KB)
We develop a simple 1D model for the scattering of an incoming particle hitting the surface of mica crystal, the transmission of energy through the crystal by a localized mode, and the ejection of atom(s) at the incident or distant face. This is the first attempt to model the experiment described by Russell and Eilbeck in 2007 (EPL, 78, 10004). Although very basic, the model shows many interesting features, for example a complicated energy dependent transition between breather modes and a kink mode, and multiple ejections at both incoming and distant surfaces. In addition, the effect of a heavier surface layer is modelled, which can lead to internal reflections of breathers or kinks at the crystal surface.
Mechanisms of recovery of radiation damage based on the interaction of quodons with crystal defects
Vladimir Dubinko
2011, 4(5): 1119-1128 doi: 10.3934/dcdss.2011.4.1119 +[Abstract](28) +[PDF](1146.7KB)
A majority of radiation effects studies are connected with creation of radiation-induced defects in the crystal bulk, which causes the observed degradation of material properties, called radiation damage. In the present paper we consider mechanisms of recovery of the radiation damage, based on the radiation-induced formation of quodons (energetic, mobile, highly localized lattice excitations that propagate great distances along close-packed crystal directions) and their interaction with crystal defects such as voids and dislocations. The rate theory of microstructure evolution in solids modified with account of quodon-induced reactions is applied for description of the radiation-induced annealing of voids observed under low temperature ion irradiation of nickel. Comparison of the theory with experimental data is used for a quantitative estimation of the propagation range of quodons in metals. Some other related phenomena in radiation physics of crystals are discussed, which include the void lattice formation and electron-plastic effect.
Travelling waves of forced discrete nonlinear Schrödinger equations
Michal Fečkan and  Vassilis M. Rothos
2011, 4(5): 1129-1145 doi: 10.3934/dcdss.2011.4.1129 +[Abstract](50) +[PDF](421.0KB)
We address the existence and bifurcation of periodic travelling wave solutions in forced spatially discrete nonlinear Schrödinger equations with local interactions. We consider polynomial type and bounded nonlinearities. The mathematical methods are based in using Palais-Smale conditions and variational methods. Some generalizations are also discussed.
Nonlinear lattice models for biopolymers: Dynamical coupling to a ionic cloud and application to actin filaments
Cynthia Ferreira , Guillaume James and  Michel Peyrard
2011, 4(5): 1147-1166 doi: 10.3934/dcdss.2011.4.1147 +[Abstract](50) +[PDF](715.3KB)
This paper is a first attempt to derive a qualitatively simple model coupling the dynamics of a charged biopolymer and its diffuse cloud of counterions. We consider here the case of a single actin filament. A zig-zag chain model introduced by Zolotaryuk et al [28] is used to represent the actin helix, and calibrated using experimental data on the stiffness constant of actin. Starting from the continuum drift-diffusion model describing counterion dynamics, we derive a discrete damped diffusion equation for the quantity of ionic charges in a one-dimensional grid along actin. The actin and ionic cloud models are coupled via electrostatic effects. Numerical simulations of the coupled system show that mechanical waves propagating along the polymer can generate charge density waves with intensities in the $pA$ range, in agreement with experimental measurements of ionic currents along actin.
Complex spatiotemporal behavior in a chain of one-way nonlinearly coupled elements
Yuri B. Gaididei , Rainer Berkemer , Carlos Gorria , Peter L. Christiansen , Atsushi Kawamoto , Takahiro Shiga , Mads P. Sørensen and  Jens Starke
2011, 4(5): 1167-1179 doi: 10.3934/dcdss.2011.4.1167 +[Abstract](46) +[PDF](204.5KB)
The dynamics of asymmetrically coupled nonlinear elements is considered. It is shown that there are two distinctive regimes of oscillatory behavior of one-way nonlinearly coupled elements depending on the relaxation time and the strength of the coupling. In the subcritical regime when the relaxation time is shorter than a critical one a spatially uniform stationary state is stable. In the supercritical regime due to a Hopf bifurcation traveling waves spontaneously create and propagate along the system. Our analytical approach is in good agreement with numerical simulations of the fully nonlinear model.
Bose-Einstein condensates and spectral properties of multicomponent nonlinear Schrödinger equations
Vladimir S. Gerdjikov
2011, 4(5): 1181-1197 doi: 10.3934/dcdss.2011.4.1181 +[Abstract](30) +[PDF](414.4KB)
We analyze the properties of the soliton solutions of a class of models describing one-dimensional BEC with spin $F$. We describe the minimal sets of scattering data which determine uniquely both the corresponding potential of the Lax operator and its scattering matrix. Next we give several reductions of these MNLS, derive their $N$-soliton solutions and analyze the soliton interactions. Finally we prove an important theorem proving that if the initial conditions satisfy the reduction then one gets a solution of the reduced MNLS.
Multiple dark solitons in Bose-Einstein condensates at finite temperatures
P.G. Kevrekidis and  Dimitri J. Frantzeskakis
2011, 4(5): 1199-1212 doi: 10.3934/dcdss.2011.4.1199 +[Abstract](52) +[PDF](536.2KB)
We study analytically, as well as numerically, single- and multiple-dark matter-wave solitons in atomic Bose-Einstein condensates at finite temperatures. Our analysis is based on the study of the dissipative Gross-Pitaevskii equation, which incorporates a phenomenological damping term accounting for the interaction of the condensate with the thermal cloud. We illustrate how the negative Krein sign eigenmodes (associated with the the single- or multiple-dark soliton states) can give rise to Hopf bifurcations and oscillatory instabilities, whose ensuing dynamics is also elucidated. In all cases, the finite-temperature induced dynamics results in soliton decay, and the system eventually asymptotes to the ground state.
Dissipative solitons in binary fluid convection
Isabel Mercader , Oriol Batiste , Arantxa Alonso and  Edgar Knobloch
2011, 4(5): 1213-1225 doi: 10.3934/dcdss.2011.4.1213 +[Abstract](38) +[PDF](3273.3KB)
A horizontal layer containing a miscible mixture of two fluids can produce dissipative solitons when heated from below. The physics of the system is described, and dissipative solitons are computed using numerical continuation for three distinct sets of experimentally realizable parameter values. The stability of the solutions is investigated using direct numerical integration in time and related to the stability properties of the competing periodic state.
Continuation and bifurcations of breathers in a finite discrete NLS equation
Panayotis Panayotaros
2011, 4(5): 1227-1245 doi: 10.3934/dcdss.2011.4.1227 +[Abstract](57) +[PDF](11244.7KB)
We present results on the continuation of breathers in the discrete cubic nonlinear Schrödinger equation in a finite one-dimensional lattice with Dirichlet boundary conditions. In the limit of small inter-site coupling the equation has a finite number of breather solutions and as we increase the coupling we see numerically that all breather branches undergo either fold or pitchfork bifurcations. We also see branches that persist for arbitrarily large coupling and converge to the linear normal modes of the system. The stability of the breathers that persist generally changes as the coupling is varied, although there are at least two branches that preserve their linear and nonlinear stability properties throughout the continuation.
Breather-mediated energy transfer in proteins
Francesco Piazza and  Yves-Henri Sanejouand
2011, 4(5): 1247-1266 doi: 10.3934/dcdss.2011.4.1247 +[Abstract](64) +[PDF](1551.8KB)
In this paper we investigate how energy is redistributed across protein structures, following localized kicks, within the framework of a nonlinear network model. We show that energy is directed most of the times to a few specific sites, systematically within the stiffest regions. This effect is sharpened as the energy of the kicks is increased, with fractions of transferred energy as high as 70% already for kicks above $20$ kcal/mol. Remarkably, we show that such site-selective, high-yield transfers mark the spontaneous formation of spatially localized, time-periodic vibrations at the target sites, acting as efficient energy-collecting centers. A comparison of our simulations with a previously developed theory reveals that such energy-pinning modes are discrete breathers, able to carry energy across the structure in an quasi-coherent fashion by jumping from site to site.
Persistent mobile lattice excitations in a crystalline insulator
Francis Michael Russell and  J. C. Eilbeck
2011, 4(5): 1267-1285 doi: 10.3934/dcdss.2011.4.1267 +[Abstract](38) +[PDF](3974.8KB)
We examine tracks in crystals of muscovite of high energy charged particles, and of mobile lattice excitations created by kinetic atomic scattering. The mobile lattice excitations are interpreted as a type of breather, here called a quodon. The typical energy of a quodon can be found from the decay of potassium K40 atoms in the crystal and supports their interpretation as a type of breather. In turn, this establishes a unique signature for energetic quodons, the 'kinked-line' tracks, allowing discrimination against tracks formed by charged particles. The stability of quodons against crystal defects and thermal motion is considered. Measurements on energetic quodon tracks, with flight paths up to 530mm, show that they can propagate more than 109 unit cells with no evidence of energy loss. This suggests that quodons might persist indefinitely in certain crystals of high quality. Evidence is presented for a new type of mobile lattice excitation that is capable of creating energetic quodons, which also is stable against lattice defects. Possible practical applications of quodons are considered briefly. Although quodons can induce fusion in deuterium or tritium, present indications are that the rate is too low to be of practical use. Finally, a nonlinear lattice effect that might increase this rate is suggested.
Logic operations demonstrated with localized vibrations in a micromechanical cantilever array
Masayuki Sato , Naoki Fujita and  A. J. Sievers
2011, 4(5): 1287-1298 doi: 10.3934/dcdss.2011.4.1287 +[Abstract](39) +[PDF](621.2KB)
A method is presented for realizing logic operations in a micro-mechanical cantilever array based on the timed application of a lattice disturbance to control the properties of intrinsic localized modes (ILMs). The application of a specific inhomogeneous field destroys a driver-locked ILM, while the same operation can create an ILM if initially no-ILM exists. Logic states "1" and "0" correspond to "present" or "absent" ILM.
Exact solutions for periodic and solitary matter waves in nonlinear lattices
Cheng Hou Tsang , Boris A. Malomed and  Kwok Wing Chow
2011, 4(5): 1299-1325 doi: 10.3934/dcdss.2011.4.1299 +[Abstract](48) +[PDF](1345.7KB)
We produce three vast classes of exact periodic and solitonic solutions to the one-dimensional Gross-Pitaevskii equation (GPE) with the pseudopotential in the form of a nonlinear lattice (NL), induced by a spatially periodic modulation of the local nonlinearity. It is well known that NLs in Bose-Einstein condensates (BECs) may be created by means of the Feshbach-resonance technique. The model may also include linear potentials with the same periodicity. The NL modulation function, the linear potential (if any), and the corresponding exact solutions are expressed in terms of the Jacobi's elliptic functions of three types, cn, dn, and sn, which give rise to the three different classes of the solutions. The potentials and associated solutions are parameterized by two free constants and an additional sign parameter in the absence of the linear potential. In the presence of the latter, the solution families feature two additional free parameters. The families include both sign-constant and sign-changing NLs. Density maxima of the solutions may coincide with either minima or maxima of the periodic pseudopotential. The solutions reduce to solitons in the limit of the infinite period. The stability of the solutions is tested via systematic direct simulations of the GPE. As a result, stability regions are identified for the periodic solutions and solitons. The periodic patterns of cn type, and the respective limit-form solutions in the form of bright solitons, may be stable both in the absence and presence of the linear potential. On the contrary, the stability of the two other solution classes, of the dn and sn types, is only possible with the linear potential.
Dark solitary waves in nonlocal nonlinear Schrödinger systems
David Usero
2011, 4(5): 1327-1340 doi: 10.3934/dcdss.2011.4.1327 +[Abstract](36) +[PDF](414.7KB)
Dark soliton-like solutions are analyzed in the context of a certain nonlocal nonlinear Schrödinger Equation with nonlocal dispersive term of Kac-Baker type. Main purpose is to investigate such solutions with negative nonlinear term and the presence of general integral dispersive terms. First the model is presented and the properties of the fundamental solution, the continuous wave, is studied. Dark solitary waves are perturbations of this plane wave. The study of dark type of solutions is divided in two different cases black and dark solitary waves. The range of existence of such solutions is studied analytically, and also their physical quantities like norm, momentum and energy. Usual behavior of nonlinear systems under nonlocal dispersive terms is found.
Snake-to-isola transition and moving solitons via symmetry-breaking in discrete optical cavities
Alexey Yulin and  Alan Champneys
2011, 4(5): 1341-1357 doi: 10.3934/dcdss.2011.4.1341 +[Abstract](32) +[PDF](1134.1KB)
This paper continues an investigation into a one-dimensional lattice equation that models the light field in a system comprised of a periodic array of pumped optical cavities with saturable nonlinearity. The additional effects of a spatial gradient of the phase of the pump field are studied, which in the presence of loss terms is shown to break the spatial reversibility of the steady problem. Unlike for continuum systems, small symmetry-breaking is argued to not lead directly to moving solitons, but there remains a pinning region in which there are infinitely many distinct stable stationary solitons of arbitrarily large width. These solitons are no-longer arranged in a homoclinic snaking bifurcation diagrams, but instead break up into discrete isolas. For large enough symmetry-breaking, the fold bifurcations of the lowest intensity solitons no longer overlap, which is argued to be the trigger point of moving localised structures. Due to the dissipative nature of the problem, any radiation shed by these structures is damped and so they appear to be true attractors. Careful direct numerical simulations reveal that branches of the moving solitons undergo unsual hysteresis with respect to the pump, for sufficiently large symmetry breaking.

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