Collective Coordinate Approach Research Articles

Lévy flights as an emergent phenomenon in a spatially extended system

Anomalous diffusion and Lévy flights, which are characterized by the occurrence of random discrete jumps of all scales, have been observed in a plethora of natural and engineered systems, ranging from the motion of molecules to climate signals. Mathematicians have recently unveiled mechanisms to generate anomalous diffusion, both stochastically and deterministically. However, there exists to the best of our knowledge no explicit example of a spatially extended system which exhibits anomalous diffusion without being explicitly driven by Lévy noise. We show here that the Landau–Lifshitz–Gilbert equation, a stochastic partial differential equation (SPDE), despite only being driven by Gaussian white noise, exhibits superdiffusive behaviour. The anomalous diffusion is an entirely emergent behaviour and manifests itself in jumps in the location of its travelling front solution. Using a collective coordinate approach, we reduce the SPDE to a set of stochastic differential equations driven by Gaussian white noise. This allows us to identify the mechanism giving rise to the anomalous diffusion as random widening events of the front interface.

Synchronization transitions in Kuramoto networks with higher-mode interaction.

Synchronization is an omnipresent collective phenomenon in nature and technology, whose understanding is still elusive for real-world systems in particular. We study the synchronization transition in a phase oscillator system with two nonvanishing Fourier-modes in the interaction function, hence going beyond the Kuramoto paradigm. We show that the transition scenarios crucially depend on the interplay of the two coupling modes. We describe the multistability induced by the presence of a second coupling mode. By extending the collective coordinate approach, we describe the emergence of various states observed in the transition from incoherence to coherence. Remarkably, our analysis suggests that, in essence, the two-mode coupling gives rise to states characterized by two independent but interacting groups of oscillators. We believe that these findings will stimulate future research on dynamical systems, including complex interaction functions beyond the Kuramoto-type.

Emergent Magnetic Field and Nonzero Gyrovector of the Toroidal Magnetic Hopfion

Magnetic hopfions are localized magnetic solitons with a nonzero 3D topological charge (Hopf index). Herein, an analytical calculation of the magnetic hopfion gyrovector is presented and it is shown that it does not vanish even in an infinite sample. The calculation method is based on the concept of the emergent magnetic field. The particular case of the simplest nontrivial toroidal hopfion with the Hopf index in the cylindrical magnetic dot is considered and dependencies of the gyrovector components on the dot sizes are calculated. Nonzero hopfion gyrovector is important in any description of the hopfion dynamics within the collective coordinate Thiele's approach.

Temperature dependence of current-driven and Brownian skyrmion dynamics in ferrimagnets with compensation point

Magnetic skyrmions are topological spin textures and promising candidates for novel spintronic applications. Recent studies on the current-driven dynamics of ferromagnetic (FM) skyrmions revealed that they exhibit an undesirable transverse motion, the skyrmion Hall effect. For antiferromagnetic (AFM) skyrmions, a vanishing skyrmion Hall effect was predicted, along with faster dynamics. However, their zero net magnetization obstructs efficient detection. Ferrimagnetic (FI) materials promise to combine both advantages: fast, AFM-like dynamics and easy read-out via stray fields. Here, we investigate the current-driven and Brownian dynamics of skyrmions in a FI with a compensation point. We perform atomistic spin dynamics simulations based on a model Hamiltonian and the stochastic Landau-Lifshitz-Gilbert equation supplemented with spin-orbit torques, accompanied by analytical calculations based on a collective coordinate approach. Our results unveil a nonmonotonic temperature dependence of the velocities and the diffusion coefficient with a strong enhancement at the angular momentum compensation temperature, due to scaling from FM- to AFM-like dynamics. These findings open up a new pathway for the efficient manipulation of skyrmion dynamics via temperature.

Walker breakdown of Brownian domain wall dynamics

The Brownian motion of domain walls in uniaxial and biaxial ferromagnetic nanowires is studied, comparing spin dynamics simulations with analytical calculations within the framework of a collective coordinate approach. Our results demonstrate that the interplay between spatial and angular dynamics gives rise to a complex time dependence of the MSD in biaxial nanowires and to a drastically reduced diffusion coefficient in uniaxial nanowires, analogous to magnetic skyrmions. This diffusion suppression is also responsible for the peculiar temperature dependence of the diffusion coefficient in biaxial systems: while it is found to scale linearly with temperature up to a certain threshold, the emergence of a Walker breakdown of Brownian motion is responsible for a reduction of the diffusion coefficient with increasing temperature above this threshold.

Collective coordinate analysis of soliton scattering by an asymmetric potential

The scattering of solitons by an asymmetric potential was investigated using a collective coordinate approach and numerical simulation. The centre of mass and width of the soliton were analyzed. A detailed comparison of theoretical predictions and results of numerical simulations of a nonlinear Schrodinger equation is presented. Unidirectional flow of soliton was also investigated numerically.

Collective coordinate framework to study solitary waves in stochastically perturbed Korteweg-de Vries equations.

Stochastically perturbed Korteweg-de Vries (KdV) equations are widely used to describe the effect of random perturbations on coherent solitary waves. We present a collective coordinate approach to describe the effect on coherent solitary waves in stochastically perturbed KdV equations. The collective coordinate approach allows one to reduce the infinite-dimensional stochastic partial differential equation (SPDE) to a finite-dimensional stochastic differential equation for the amplitude, width and location of the solitary wave. The reduction provides a remarkably good quantitative description of the shape of the solitary waves and its location. Moreover, the collective coordinate framework can be used to estimate the timescale of validity of stochastically perturbed KdV equations for which they can be used to describe coherent solitary waves. We describe loss of coherence by blow-up as well as by radiation into linear waves. We corroborate our analytical results with numerical simulations of the full SPDE.

Mesoscopic model reduction for the collective dynamics of sparse coupled oscillator networks.

The behavior at bifurcation from global synchronization to partial synchronization in finite networks of coupled oscillators is a complex phenomenon, involving the intricate dynamics of one or more oscillators with the remaining synchronized oscillators. This is not captured well by standard macroscopic model reduction techniques that capture only the collective behavior of synchronized oscillators in the thermodynamic limit. We introduce two mesoscopic model reductions for finite sparse networks of coupled oscillators to quantitatively capture the dynamics close to bifurcation from global to partial synchronization. Our model reduction builds upon the method of collective coordinates. We first show that standard collective coordinate reduction has difficulties capturing this bifurcation. We identify a particular topological structure at bifurcation consisting of a main synchronized cluster, the oscillator that desynchronizes at bifurcation, and an intermediary node connecting them. Utilizing this structure and ensemble averages, we derive an analytic expression for the mismatch between the true bifurcation from global to partial synchronization and its estimate calculated via the collective coordinate approach. This allows to calibrate the standard collective coordinate approach without prior knowledge of which node will desynchronize. We introduce a second mesoscopic reduction, utilizing the same particular topological structure, which allows for a quantitative dynamical description of the phases near bifurcation. The mesoscopic reductions significantly reduce the computational complexity of the collective coordinate approach, reducing from O(N2) to O(1). We perform numerical simulations for Erdős-Rényi networks and for modified Barabási-Albert networks demonstrating remarkable quantitative agreement at and close to bifurcation.

Domain-Wall Damping in Ultrathin Nanostripes with Dzyaloshinskii-Moriya Interaction

Asymmetrically sandwiched thin magnetic layers with perpendicular anisotropy and Dzyaloshinskii-Moriya interaction (DMI) is the prospective material science platform for spin-orbitronic technologies that rely on the motion of chiral magnetic textures, like skyrmions or chiral domain walls (DWs). The dynamic performance of a DW-based racetrack is defined by the strength of the DMI and the DW damping. The determination of the latter parameter is typically done based on technically challenging DW motion experiments. Here, we propose a method to access both the DMI constant and DW damping from static experiments by monitoring the tilt of magnetic DWs in nanostripes. We experimentally demonstrate that in perpendicularly magnetized ${\mathrm{//Cr}\mathrm{O}}_{x}\mathrm{/Co/Pt}$ stacks, DWs can be trapped on edge roughness in a metastable tilted state as a result of the DW dynamics driven by an external magnetic field. The measured tilt can be correlated to the DMI strength and DW damping in a self-consistent way in the frame of a theoretical formalism based on the collective coordinate approach.

Analytical model of the deformation-induced inertial dynamics of a magnetic vortex

We present an analytical model to account for the deformation-induced inertial dynamics of a magnetic vortex. The model is based on a deformation of the vortex core profile based on the Döring kinetic field, whereby the deformation amplitudes are promoted to dynamical variables in a collective-coordinate approach that provides a natural extension to the Thiele model. This extended model describes complex transients due to inertial effects and the variation of the effective mass with velocity. The model also provides a quantitative description of the inertial dynamics leading up to vortex core reversal, which is analogous to the Walker transition in domain wall dynamics. Our work paves the way for a standard prescription for describing the inertial effects of topological magnetic solitons.

Model reduction for the collective dynamics of globally coupled oscillators: From finite networks to the thermodynamic limit.

Model reduction techniques have been widely used to study the collective behavior of globally coupled oscillators. However, most approaches assume that there are infinitely many oscillators. Here, we propose a new ansatz, based on the collective coordinate approach, that reproduces the collective dynamics of the Kuramoto model for finite networks to high accuracy, yields the same bifurcation structure in the thermodynamic limit of infinitely many oscillators as previous approaches, and additionally captures the dynamics of the order parameter in the thermodynamic limit, including critical slowing down that results from a cascade of saddle-node bifurcations.

Model reduction for the Kuramoto-Sakaguchi model: The importance of nonentrained rogue oscillators.

The Kuramoto-Sakaguchi model for coupled phase oscillators with phase frustration is often studied in the thermodynamic limit of infinitely many oscillators. Here we extend a model reduction method based on collective coordinates to capture the collective dynamics of finite-size Kuramoto-Sakaguchi models. We find that the inclusion of the effects of rogue oscillators is essential to obtain an accurate description, in contrast to the original Kuramoto model, where we show that their effects can be ignored. We further introduce a more accurate ansatz function to describe the shape of synchronized oscillators. Our results from this extended collective coordinate approach reduce in the thermodynamic limit to the well-known mean-field consistency relations. For finite networks we show that our model reduction describes the collective behavior accurately, reproducing the order parameter, the mean frequency of the synchronized cluster, and the size of the cluster at a given coupling strength, as well as the critical coupling strength for partial and for global synchronization.

Dynamics of distorted and undistorted soliton molecules in a mode-locked fiber laser

Recent developments in real-time ultrafast measurement techniques have enabled us to prove experimentally that soliton molecules execute internal motions with some aspects similar to those of a matter molecule. Such an analogy between the dynamics of soliton molecules and the dynamics of matter molecules is based on the assumption that the dissipative solitons constituting a molecule are rigid entities sharing a common profile. Whereas this assumption drastically reduces the number of degrees of freedom, it does not hold true in general and we demonstrate that it overlooks some of the essential dynamical features of the soliton molecule. We present a theoretical study based on the principle that the different pulse constituents of a soliton molecule are deformable entities. Specifically, by using of a collective coordinate approach to investigate bi- and trisoliton molecules, we reveal features such as symmetric or asymmetric distortions of their profiles, and energy exchange processes between them. This implies, in subsequent experiments, the use of characterization techniques that can be used to retrieve a larger number of degrees of freedom.

Variational approximations of soliton dynamics in the Ablowitz-Musslimani nonlinear Schrödinger equation

We study the integrable nonlocal nonlinear Schrödinger equation proposed by Ablowitz and Musslimani, that is considered as a particular example of equations with parity-time (PT) symmetric self-induced potential. We consider dynamics (including collisions) of moving solitons. Analytically we develop a collective coordinate approach based on variational methods and examine its applicability in the system. We show numerically that a single moving soliton can pass the origin and decays or be trapped at the origin and blows up at a finite time. Using a standard soliton ansatz, the variational approximation can capture the dynamics well, including the finite-time blow up, even though the ansatz is relatively far from the actual blowing-up soliton solution. In the case of two solitons moving towards each other, we show that there can be a mass transfer between them, in addition to wave scattering. We also demonstrate that defocusing nonlinearity can support bright solitons.

A collective coordinate framework to study the dynamics of travelling waves in stochastic partial differential equations

We propose a formal framework based on collective coordinates to reduce infinite-dimensional stochastic partial differential equations (SPDEs) with symmetry to a set of finite-dimensional stochastic differential equations which describe the shape of the solution and the dynamics along the symmetry group. We study SPDEs arising in population dynamics with multiplicative noise and additive symmetry breaking noise. The collective coordinate approach provides a remarkably good quantitative description of the shape of the travelling front as well as its diffusive behaviour, which would otherwise only be available through costly computational experiments. We corroborate our analytical results with numerical simulations of the full SPDE.

Skyrmion Gas Manipulation for Probabilistic Computing

The topologically protected magnetic spin configurations known as skyrmions offer promising applications due to their stability, mobility and localization. In this work, we emphasize how to leverage the thermally driven dynamics of an ensemble of such particles to perform computing tasks. We propose a device employing a skyrmion gas to reshuffle a random signal into an uncorrelated copy of itself. This is demonstrated by modelling the ensemble dynamics in a collective coordinate approach where skyrmion-skyrmion and skyrmion-boundary interactions are accounted for phenomenologically. Our numerical results are used to develop a proof-of-concept for an energy efficient ($\sim\mu\mathrm{W}$) device with a low area imprint ($\sim\mu\mathrm{m}^2$). Whereas its immediate application to stochastic computing circuit designs will be made apparent, we argue that its basic functionality, reminiscent of an integrate-and-fire neuron, qualifies it as a novel bio-inspired building block.

Elliptic-type soliton combs in optical ring microresonators

Soliton crystals are periodic patterns of multi-spot optical fields formed from either time or space entanglements of equally separated identical high-intensity pulses. These specific nonlinear optical structures have gained interest in recent years with the advent and progress in nonlinear optical fibers and fiber lasers, photonic crystals, wave-guided wave systems and most recently optical ring microresonator devices. In this work an extensive analysis of characteristic features of soliton crystals is carried out, with emphasis on their one-to-one correspondance with Elliptic solitons. In this purpose we examine their formation, their stability and their dynamics in ring-shaped nonlinear optical media within the framework of the Lugiato-Lefever equation. The stability analysis deals with internal modes of the system via a $2\times2$-matrix Lam\'e type eigenvalue problem, the spectrum of which is shown to possess a rich set of boundstates consisting of stable zero-fequency modes and unstable decaying as well as growing modes. Turning towards the dynamics of Elliptic solitons in ring-shaped fiber resonators with Kerr nonlinearity, first of all we propose a collective-coordinate approach, based on a Lagrangian formalism suitable for Elliptic-soliton solutions to the nonlinear Schr\'odinger equation with an arbitrary perturbation. Next we derive time evolutions of Elliptic-soliton parameters in the specific context of ring-shaped optical fiber resonators, where the optical field evolution is tought to be governed by the Lugiato-Lefever equation. By solving numerically the collective-coordinate equations an analysis of the amplitude, the position, the phase of internal oscillations, the phase velocity and the energy is carried out and reveals a complex dynamics of the Elliptic soliton in ring-shaped optical microresonators.

Resonant kink–antikink scattering through quasinormal modes

We investigate the role that quasinormal modes can play in kink–antikink collisions, via an example based on a deformation of the ϕ4 model. We find that narrow quasinormal modes can store energy during collision processes and later return it to the translational degrees of freedom. Quasinormal modes also decay, which leads to energy leakage, causing a closing of resonance windows and an increase of the critical velocity. We observe similar phenomena in an effective model, a small modification of the collective-coordinate approach to the ϕ4 model.

Pulsating Solitons in the Two-Dimensional Complex Swift-Hohenberg Equation

In this paper, we performed an investigation of the dissipative solitons of the two-dimensional (2D) Complex Swift-Hohenberg equation (CSHE). Stationary to pulsating soliton bifurcation analysis of the 2D CSHE is displayed. The approach is based on the semi-analytical method of collective coordinate approach. This method is constructed on a reduction from an infinite-dimensional dynamical dissipative system to a finite-dimensional model. The reduced model helps to obtain approximately the boundaries between the stationary and pulsating solutions. We analyzed the dynamics and characteristics of the pulsating solitons. Then we obtained the bifurcation diagram for a definite range of the saturation of the Kerr nonlinearity values. This diagram reveals the effect of the saturation of the Kerr nonlinearity on the period pulsations. The results show that when the parameter of the saturation of the Kerr nonlinearity increases, one period pulsating soliton solution bifurcates to double period pulsations.

Implementation of one-dimensional domain wall dynamics simulator

We implemented a one-dimensional domain wall (DW) dynamics simulator based on the well-developed collective coordinate approach to demonstrate DW motion under a given magnetic field and/or current flow. The simulator adopted all known influences, including three-dimensional external magnetic fields, spin transfer torque with non-adiabatic contribution, spin Hall effect, Rashba effect, and Dzyaloshinskii-Moriya interaction. The simulator can calculate the position, velocity, internal magnetization angle, and tilting angle of the domain wall to the current direction or wire axis under given simulation conditions and material parameters. It will not only provide physical insights of domain wall dynamics to experimentalists, but also can be used to more easily simulate various physical circumstances before running time-consuming micromagnetic simulations or real experiments.