Collective Coordinate Approach Research Articles

Variational approximation and the use of collective coordinates

We consider propagating, spatially localized waves in a class of equations that contain variational and nonvariational terms. The dynamics of the waves is analyzed through a collective coordinate approach. Motivated by the variational approximation, we show that there is a natural choice of projection onto collective variables for reducing the governing (nonlinear) partial differential equation (PDE) to coupled ordinary differential equations (ODEs). This projection produces ODEs whose solutions are exactly the stationary states of the effective Lagrangian that would be considered in applying the variational approximation method. We illustrate our approach by applying it to a modified Fisher equation for a traveling front, containing a non-constant-coefficient nonlinear term. We present numerical results that show that our proposed projection captures both the equilibria and the dynamics of the PDE much more closely than previously proposed projections.

Collective coordinate theory for light pulses in fibers: The reduced projection operators

Abstract We consider optical waveguides in which light pulses may execute complex dynamical behaviors including translational motions accompanied with strong internal vibrations. Such systems necessarily generate various types of collective motion, in which each collective mode is describable by a collective coordinate. We present a novel projection operator formalism for deriving the equations of motion of the collective coordinates and coupled fields. This formalism is built up by treating separately the dynamics of the pulse phase and that of its amplitude, that is, by using two distinct projection operators (one for the amplitude, and the other one for the phase). This new pair of operators, which we call reduced projection operators , has as main virtue of having dimension reduced by half as compared to that of the conventional operators that includes both pulse amplitude and phase together. The main interest of the reduced projection operators lies in the ease with which the equations of motion are derived when compared with the amount of algebra needed to obtain the same equations from the conventional projection operators. We provide examples of concrete situations that illustrate the effectiveness of the collective-coordinate approach based on the reduced projection operators.

Scattering of a matter-wave single soliton and a two-soliton molecule by an attractive potential

Scattering of a matter-wave single soliton and two-soliton molecule incident on the modified Pöschl-Teller potential well has been studied by means of a collective coordinate approach and numerical simulations of the Gross-Pitaevskii equation. Despite the attractive nature of the potential we observe total reflection of solitons in particular ranges of parameters, which is the signature of quantum behavior displayed by the matter-wave soliton. For other particular sets of parameters unscathed transmission of solitons and molecules through the potential well has been identified. A specific feature of this process is that the soliton passing through the potential well overtakes the freely propagating counterpart; i.e., its mean position appears to have been advanced in time. An array of such potentials makes the "time advance" effect even more pronounced, so that scattered solitons move well ahead of nonscattered ones, fully preserving their initial shape and velocity. A possible application of the obtained results is pointed out.

Interacting Wave Fronts and Rarefaction Waves in a Second Order Model of Nonlinear Thermoviscous Fluids

A wave equation including nonlinear terms up to the second order for a thermoviscous Newtonian fluid is proposed. In the lossless case this equation results from an expansion to third order of the Lagrangian for the fundamental non-dissipative fluid dynamical equations. Thus it preserves the Hamiltonian structure, in contrast to the Kuznetsov equation, a model often used in nonlinear acoustics. An exact traveling wave front solution is derived from a generalized traveling wave assumption for the velocity potential. Numerical studies of the evolution of a number of arbitrary initial conditions as well as head-on colliding and confluent wave fronts exhibit several nonlinear interaction phenomena. These include wave fronts of changed velocity and amplitude along with the emergence of rarefaction waves. An analysis using the continuity of the solutions as well as the boundary conditions is proposed. The dynamics of the rarefaction wave is approximated by a collective coordinate approach in the energy balance equation.

Resonant trapping in the transport of a matter-wave soliton through a quantum well

We theoretically investigate the scattering of bright solitons in a Bose-Einstein condensate on narrow attractive potential wells. Reflection, transmission, and trapping of an incident soliton are predicted to occur with remarkably abrupt transitions upon varying the potential depth. Numerical simulations of the nonlinear Schr\"odinger equation are complemented by a variational collective coordinate approach. The mechanism for nonlinear trapping is found to rely both on resonant interaction between the soliton and bound states in the potential well and on the radiation of small-amplitude waves. These results suggest that solitons can be used to probe bound states that are not accessible through scattering with single atoms.

Domain Wall Dynamics under an In-Plane Rotating Magnetic Field in a Nanowire with Perpendicular Magnetic Anisotropy

As an alternative to current or static magnetic field driven domain wall (DW) motion, we studied the dynamics of DW motion under an in-plane rotating magnetic field (IRMF) in a metallic nanowire with a perpendicular magnetic anisotropy. An equation describing the DW motion was obtained with a one-dimensional analytical model based on a collective coordinate approach. The DW velocity can easily be controlled up to hundreds of meters per second by varying the IRMF amplitude and frequency. The validity of the equation for DW motion was confirmed with micromagnetic simulations.

Breather-breather interactions in sine-Gordon systems using collective coordinate approach

We investigate the interaction between breathers in sine-Gordon systems using a collective coordinate approach. Focusing on the in-phase or out-of-phase oscillation mode of two identical breathers, we derive a simple ordinary differential equation for the collective coordinate: the separation between the centers of mass of the two breathers R. Analytic solutions of this equation can reproduce quantitatively the results of a direct numerical simulation of the sine-Gordon equation over the whole parameter range. The interaction between the two breathers is attractive (repulsive) for the in-phase (out-of-phase) oscillation mode with an asymptotic exponential dependence on R. We find that the interaction within the innermost kink-antikink (kink-kink) pair for the in-phase (out-of-phase) case plays the most significant role in determining the sign and the strength of the effective breather-breather interaction. We also find an internal oscillation mode of the in-phase oscillating breather pair and obtain an analytic expression for the angular frequency of the mode.

Soliton ratchets in sine-Gordon systems with additive inhomogeneities

We investigate the ratchet dynamics of solitons of a sine-Gordon system with additive inhomogeneities. We show by means of a collective coordinate approach that the soliton moves like a particle in an effective potential which is a result of the inhomogeneities. Different degrees of freedom of the soliton are used as collective coordinates in order to study their influence on the motion of the soliton. The collective coordinates considered are the soliton position, its width and offset, and the height of the spikes that appear on the soliton. The results of the theory are compared with numerical simulations of the full system.

Concept of the field-driven domain wall motion memory

In this paper, the concept of field-driven domain wall motion memory is presented. It is confirmed that a domain is shifted with a carefully designed non-uniform field by micromagnetic simulations. The shift of a domain—a bit—can be established by the motion of two domain walls to the same direction and the same distance. In order to get a better understanding of the domain wall motion under the non-uniform transverse magnetic field, we investigate the motion of the transverse Néel-type domain wall by micromagnetic simulations and the collective coordinate approach. The validity of the equation of motion for the domain wall is confirmed by the micromagnetic simulations as functions of the gradient of the non-uniform field, the saturation magnetization, and the Gilbert damping parameter α.

Dynamics of a vortex domain wall in a magnetic nanostrip: Application of the collective-coordinate approach

The motion of a vortex domain wall in a ferromagnetic strip of submicron width under the influence of an external magnetic field exhibits three distinct dynamical regimes. In a viscous regime at low fields the wall moves rigidly with a velocity proportional to the field. Above a critical field the viscous motion breaks down, giving way to oscillations accompanied by a slow drift of the wall. At still higher fields the drift velocity starts rising with the field again but with a much lower mobility $dv/dH$ than in the viscous regime. To describe the dynamics of the wall, we use the method of collective coordinates that focuses on soft modes of the system. By retaining two soft modes, parametrized by the coordinates of the vortex core, we obtain a simple description of the wall dynamics at low and intermediate applied fields that applies to both the viscous and oscillatory regimes below and above the breakdown. The calculated dynamics agrees well with micromagnetic simulations at low and intermediate values of the driving field. In higher fields, additional modes become soft and the two-mode approximation is no longer sufficient. We explain some of the significant features of vortex-domain-wall motion in high fields through the inclusion of additional modes associated with the half antivortices on the strip edge.

Equation of motion for a domain wall movement under a nonuniform transverse magnetic field

We investigate the dynamics of the domain wall in a nonuniform field driven domain wall motion memory concept. The equation of motion for a transverse Néel-type domain wall under a nonuniform transverse magnetic field is obtained with a collective coordinate approach. The validity of the equation of the motion is confirmed with micromagnetic simulations. We find that the domain wall velocity depends on the domain wall width, Gilbert damping parameter α, saturation magnetization, and the gradient of the field. The domain wall velocity of ∼100m∕s is obtained with typical material as Permalloy with a moderate field gradient (100Oe∕μm). It has promising results for memory applications.

Current-Induced Magnetic Domain-Wall Motion by Spin Transfer Torque: Collective Coordinate Approach with Domain-Wall Width Variation

The spin transfer torque generated by a spin-polarized current can induce the shift of the magnetic domain-wall position. In this work, we study theoretically the current-induced domain-wall motion by using the collective coordinate approach [Gen Tatara and Hiroshi Kohno, Phys. Rev. Lett. 92, 86601 (2004)]. The approach is extended to include not only the domain-wall position and the polarization angle changes but also the domain-wall width variation. It is demonstrated that the width variation affects the critical current.

Optimization of soliton ratchets in inhomogeneous sine-Gordon systems

Unidirectional motion of solitons can take place, although the applied force has zero average in time, when the spatial symmetry is broken by introducing a potential V(x) , which consists of periodically repeated cells with each cell containing an asymmetric array of strongly localized inhomogeneities at positions xi. A collective coordinate approach shows that the positions, heights, and widths of the inhomogeneities (in that order) are the crucial parameters so as to obtain an optimal effective potential Uopt that yields a maximal average soliton velocity. Uopt essentially exhibits two features: double peaks consisting of a positive and a negative peak, and long flat regions between the double peaks. Such a potential can be obtained by choosing inhomogeneities with opposite signs (e.g., microresistors and microshorts in the case of long Josephson junctions) that are positioned close to each other, while the distance between each peak pair is rather large. These results of the collective variable theory are confirmed by full simulations for the inhomogeneous sine-Gordon system.

Quantum collective QCD string dynamics

The string breaking model of particle production is extended in order to help explain the transverse momentum distribution in elementary collisions. Inspired by an idea of Bialas, we treat the string using a collective coordinate approach. This leads to a chromo-electric field strength which fluctuates, and in turn implies that quarks are produced according to a thermal distribution.

Ratchet effect in a damped sine-Gordon system with additive and parametric ac driving forces

We study in detail the damped sine-Gordon equation, driven by two ac forces (one is added as a parametric perturbation and the other one in an additive way), as an example of soliton ratchets. By means of a collective coordinate approach we derive an analytical expression for the average velocity of the soliton, which allows us to show that this mechanism of transport requires certain relationships both between the frequencies and between the initial phases of the two ac forces. The control of the velocity by the damping coefficient and parameters of the ac forces is also presented and discussed. All these results are subsequently checked by means of simulations for the driven and damped sine-Gordon equation that we have studied.

Simulating nanoscale functional motions of biomolecules

We are describing efficient dynamics simulation methods for the characterization of functional motion of biomolecules on the nanometer scale. Multivariate statistical methods are widely used to extract and enhance functional collective motions from molecular dynamics (MD) simulations. A dimension reduction in MD is often realized through a principal component analysis (PCA) or a singular value decomposition (SVD) of the trajectory. Normal mode analysis (NMA) is a related collective coordinate space approach, which involves the decomposition of the motion into vibration modes based on an elastic model. Using the myosin motor protein as an example we describe a hybrid technique termed amplified collective motions (ACM) that enhances sampling of conformational space through a combination of normal modes with atomic level MD. Unfortunately, the forced orthogonalization of modes in collective coordinate space leads to complex dependencies that are not necessarily consistent with the symmetry of biological macromolecules and assemblies. In many biological molecules, such as HIV-1 protease, reflective or rotational symmetries are present that are broken using standard orthogonal basis functions. We present a method to compute the plane of reflective symmetry or the axis of rotational symmetry from the trajectory frames. Moreover, we develop an SVD that best approximates the given trajectory while respecting the symmetry. Finally, we describe a local feature analysis (LFA) to construct a topographic representation of functional dynamics in terms of local features. The LFA representations are low-dimensional, and provide a reduced basis set for collective motions, but unlike global collective modes they are sparsely distributed and spatially localized. This yields a more reliable assignment of essential dynamics modes across different MD time windows.

Soliton ratchets in homogeneous nonlinear Klein-Gordon systems

We study in detail the ratchetlike dynamics of topological solitons in homogeneous nonlinear Klein-Gordon systems driven by a biharmonic force. By using a collective coordinate approach with two degrees of freedom, namely the center of the soliton, X(t), and its width, l(t), we show, first, that energy is inhomogeneously pumped into the system, generating as result a directed motion; and, second, that the breaking of the time shift symmetry gives rise to a resonance mechanism that takes place whenever the width l(t) oscillates with at least one frequency of the external ac force. In addition, we show that for the appearance of soliton ratchets, it is also necessary to break the time-reversal symmetry. We analyze in detail the effects of dissipation in the system, calculating the average velocity of the soliton as a function of the ac force and the damping. We find current reversal phenomena depending on the parameter choice and discuss the important role played by the phases of the ac force. Our analytical calculations are confirmed by numerical simulations of the full partial differential equations of the sine-Gordon and phi4 systems, which are seen to exhibit the same qualitative behavior. Our results show features similar to those obtained in recent experimental work on dissipation induced symmetry breaking.

Length scale competition in nonlinear Klein—Gordon models: A collective coordinate approach

Working within the framework of nonlinear Klein-Gordon models as a paradigmatic example, we show that length scale competition, an instability of solitons subjected to perturbations of an specific length, can be understood by means of a collective coordinate approach in terms of soliton position and width. As a consequence, we provide a natural explanation of the phenomenon in much simpler terms than any previous treatment of the problem. Our technique allows us to study the existence of length scale competition in most soliton bearing nonlinear models and can be extended to coherent structures with more degrees of freedom.

Microwave-induced fluxon bunching in weakly coupled Josephson junctions

We consider a pair of weakly coupled long Josephson junctions and investigate the conditions under which an external microwave field may phase-lock a single shuttling fluxon in each junction, and subsequently collapse the mutually repulsive fluxons into a phase-locked bunched state. Based on the coupled sine-Gordon model and the collective coordinate perturbation approach, we develop specific conditions for the physical parameters necessary to ensure the bunching of two fluxons in different junctions. The perturbation results are verified by direct numerical simulations.

RATCHETS IN HOMOGENEOUS EXTENDED SYSTEMS: INTERNAL MODES AND THE ROLE OF NOISE

We revisit the issue of directed motion induced by zero average forces in extended systems driven by ac forces. It has been shown recently that a directed energy current appears if the ac external force, f(t), breaks the symmetry f(t)=-f(t+T/2), T being the period, if topological solitons (kinks) existed in the system. In this work, a collective coordinate approach allows us to identify the mechanism through which the width oscillation drives the kink and its relation with the mathematical symmetry conditions. Furthermore, our theory predicts, and numerical simulations confirm, that the direction of motion depends on the initial phase of the driving, while the system behaves in a ratchet-like fashion if averaging over initial conditions. Finally, the presence of noise overimposed to the ac driving does not destroy the directed motion; on the contrary, it gives rise to an activation process that increases the velocity of the motion. We conjecture that this could be a signature of resonant phenomena at larger noises.