Collective Coordinate Approximation Research Articles

Collective coordinates for the hybrid model

In this work, we investigate the soliton scattering problem for anti-kink/kink and kink/anti-kink configurations. To this end, we employ two distinct methods, full simulation and collective coordinates, and compare them. We apply the collective coordinate approximation method to describe both scattering configurations in the hybrid model and show that, similar to the polynomial models [Formula: see text] and [Formula: see text], the method has limitations with respect to the initial scattering velocities. In such a way that, for certain initial velocities, the solution of collective coordinates agrees with the full simulation, and for other velocities, there is a discrepancy in the solutions obtained by these two methods. We investigate the time evolution for kink/anti-kink and anti-kink/kink scattering in the hybrid model, taking into consideration the null-vector problem. As an appendix, we performed the same inspection for kink/anti-kink scattering in the [Formula: see text] model.

Stability of exact solutions of the (2 + 1)-dimensional nonlinear Schrödinger equation with arbitrary nonlinearity parameter κ

In this work, we consider the nonlinear Schrödinger equation (NLSE) in 2+1 dimensions with arbitrary nonlinearity exponent κ in the presence of an external confining potential. Exact solutions to the system are constructed, and their stability as we increase the ‘mass’ (i.e., the L 2 norm) and the nonlinearity parameter κ is explored. We observe both theoretically and numerically that the presence of the confining potential leads to wider domains of stability over the parameter space compared to the unconfined case. Our analysis suggests the existence of a stable regime of solutions for all κ as long as their mass is less than a critical value M*(κ). Furthermore, we find that there are two different critical masses, one corresponding to width perturbations and the other one to translational perturbations. The results of Derrick’s theorem are also obtained by studying the small amplitude regime of a four-parameter collective coordinate (4CC) approximation. A numerical stability analysis of the NLSE shows that the instability curve M*(κ) versus κ lies below the two curves found by Derrick’s theorem and the 4CC approximation. In the absence of the external potential, κ = 1 demarcates the separation between the blowup regime and the stable regime. In this 4CC approximation, for κ < 1, when the mass is above the critical mass for the translational instability, quite complicated motions of the collective coordinates are possible. Energy conservation prevents the blowup of the solution as well as confines the center of the solution to a finite spatial domain. We call this regime the ‘frustrated’ blowup regime and give some illustrations. In an appendix, we show how to extend these results to arbitrary initial ground state solution data and arbitrary spatial dimension d.

Behavior of solitary waves of coupled nonlinear Schrödinger equations subjected to complex external periodic potentials with odd- symmetry

We discuss the response of both moving and trapped solitary wave solutions of a two-component nonlinear Schrödinger system in 1 + 1 dimensions to an odd- external periodic complex potential. The dynamical behavior of perturbed solitary waves is explored by conducting numerical simulations of the nonlinear system and using a collective coordinate variational approximation. We present case examples corresponding to choices of parameter values and initial conditions involved therein. The results of the collective coordinate approximation are compared against numerical simulations where we observe qualitatively good agreement between the two. Unlike the case for a single-component solitary wave in a complex periodic -symmetric potential, the collective coordinate equations do not have a small oscillation regime, and initially the height of the two components changes in opposite directions often causing instability. We find that the dynamic stability criteria we have used in the one-component case are a good indicator for the onset of dynamic instabilities in the present setup.

Parametrically driven nonlinear Dirac equation with arbitrary nonlinearity

The damped and parametrically driven nonlinear Dirac equation with arbitrary nonlinearity parameter is analyzed, when the external force is periodic in space and given by , both numerically and in a variational approximation using five collective coordinates (time dependent shape parameters of the wave function). Our variational approximation satisfies exactly the low-order moment equations. Because of competition between the spatial period of the external force , and the soliton width ls, which is a function of the nonlinearity as well as the initial frequency of the solitary wave, there is a transition (at fixed ) from trapped to unbound behavior of the soliton, which depends on the parameters r and K of the external force and the nonlinearity parameter . We previously studied this phenomena when (Quintero et al 2019 J. Phys. A: Math. Theor. 52 285201) where we showed that for the soliton oscillates in an effective potential, while for it moves uniformly as a free particle. In this paper we focus on the dependence of the transition from oscillatory to particle behavior and explicitly compare the curves of the transition regime found in the collective coordinate approximation as a function of r and K when at fixed value of the frequency . Since the solitary wave gets narrower for fixed as a function of , we expect and indeed find that the regime where the solitary wave is trapped is extended as we increase .

Exact solutions of a generalized variant of the derivative nonlinear Schrödinger equation in a Scarff II external potential and their stability properties

We obtain exact solitary wave solutions of a variant of the generalized derivative nonlinear Schrödinger equation in 1+1 dimensions with arbitrary values of the nonlinearity parameter κ in a Scarf-II potential. This variant of the usual derivative nonlinear Schrödinger equation has the properties that for real external potentials, the dynamics is derivable from a Lagrangian. The solitary wave and trapped solutions have the same form as those of the usual derivative nonlinear Schrödinger equation. We show that the solitary wave solutions are orbitally stable for . We find new exact nodeless solutions to the bound states in the external complex potential which are related to the static solutions of the equation. We also use a collective coordinate approximation to analyze the stability of the trapped solutions when the external potential is real.

Magnetic Skyrmionic Polarons.

We study a two-dimensional electron gas exchange coupled to a system of classical magnetic ions. For large Rashba spin-orbit coupling, a single electron can become self-trapped in a skyrmion spin texture self-induced in the magnetic ions system. This new quasiparticle carries electrical and topological charge as well as a large spin, and we named it as magnetic skyrmionic polaron. We study a range of parameters; temperature, exchange coupling, Rashba coupling, and magnetic field, for which the magnetic skyrmionic polaron is the fundamental state in the system. The dynamics of this quasiparticle is studied using the collective coordinate approximation, and we obtain that in the presence of an electric field the new quasiparticle shows, due to the chirality of the skyrmion, a Hall effect. Finally, we argue that the magnetic skyrmionic polarons can be found in large Rashba spin-orbit coupling semiconductors as GeMnTe.

Topological defects with power-law tails

We study interactions of kinks and antikinks of the (1 + 1)-dimensional ϕ8 model. In this model, there are kinks with mixed tail asymptotics: power-law behavior at one infinity versus exponential decay towards the other. We show that if a kink and an antikink face each other in way such that their power-law tails determine the kink-antikink interaction, then the force of their interaction decays slowly, as some negative power of distance between them. We estimate the force numerically using the collective coordinate approximation, and analytically via Manton’s method (making use of formulas derived for the kink and antikink tail asymptotics).

Collective coordinates in one-dimensional soliton models revisited

We compare numerical solutions to the full field equations to simplified approaches based on implementing three collective coordinates for kink-antikink interactions within the $\varphi^4$ and $\phi^6$ models in one time and one space dimensions. We particularly pursue the question whether the collective coordinate approximation substantiates the conjecture that vibrational modes are important for resonance structures to occur in kink-antikink scattering.

Collective coordinate approximation to the scattering of solitons in modified NLS and sine-Gordon models

We investigate the validity of collective coordinate approximations to the scattering of two solitons in several classes of (1+1) dimensional field theory models. We consider models which are deformations of the sine-Gordon (SG) or the nonlinear Schrodinger (NLS) model which posses soliton solutions (which are topological (SG) or non-topological (NLS)). Our deformations preserve their topology (SG), but change their integrability properties, either completely or partially (models become ‘quasi-integrable’). As the collective coordinate approximation does not allow for the radiation of energy out of a system we look, in some detail, at how the approximation fares in models which are ‘quasi-integrable’ and therefore have asymptotically conserved charges (i.e. charges Q(t) for which Q(t → −∞) = Q(t → ∞)). We find that our collective coordinate approximation, based on geodesic motion etc, works amazingly well in all cases where it is expected to work. This is true for the physical properties of the solitons and even for their quasi-conserved (or not) charges. The only time the approximation is not very reliable (and even then the qualitative features are reasonable, but some details are not reproduced well) involves the processes when the solitons come very close together (within one width of each other) during their scattering.

English

In this paper, we investigate the scattering of two $\phi^6$ kinks and derive the real dynamics by solving the appropriate field equation numerically employing a Runga-Kutta method. We also use a collective coordinate approximation to find approximate dynamics, with the objective being to compare the approximate motion to the real dynamics in order to test the validity of this kind of approximation, which is used extensively in the study of solitons.

Collective coordinate approximation to the scattering of solitons in the (1+1) dimensional NLS model

We present a collective coordinate approximation to model the dynamics of two interacting nonlinear Schrödinger solitons. We discuss the accuracy of this approximation by comparing our results with those of the full numerical simulations and find that the approximation is remarkably accurate when the solitons are some distance apart, and quite reasonable also during their interaction.

Kink interactions in the(1+1)-dimensionalφ6model

We study kink scattering processes in the (1+1)-dimensional $\varphi^6$ model in the framework of the collective coordinate approximation. We find critical values of the initial velocities of the colliding kinks. These critical velocities distinguish different regimes of collisions. The exact equation of motion for the $\varphi^6$ model is also solved numerically with the same initial conditions. We discuss advantages and disadvantages of the collective coordinate approximation, and also outline its applicability limits. Resonance phenomena and the so-called escape windows are also observed in the kink collisions.

Nonlinear Dirac equation solitary waves in external fields

We consider nonlinear Dirac equations (NLDE's) in the 1+1 dimension with scalar-scalar self-interaction g2/κ+1(Ψ[over ¯]Ψ)κ+1 in the presence of various external electromagnetic fields. We find exact solutions for special external fields and we study the behavior of solitary-wave solutions to the NLDE in the presence of a wide variety of fields in a variational approximation depending on collective coordinates which allows the position, width, and phase of these waves to vary in time. We find that in this approximation the position q(t) of the center of the solitary wave obeys the usual behavior of a relativistic point particle in an external field. For time-independent external fields, we find that the energy of the solitary wave is conserved but not the momentum, which becomes a function of time. We postulate that, similarly to the nonlinear Schrödinger equation (NLSE), a sufficient dynamical condition for instability to arise is that dP(t)/dq[over ̇](t)<0. Here P(t) is the momentum of the solitary wave, and q[over ̇] is the velocity of the center of the wave in the collective coordinate approximation. We found for our choices of external potentials that we always have dP(t)/dq[over ̇](t)>0, so, when instabilities do occur, they are due to a different source. We investigate the accuracy of our variational approximation using numerical simulations of the NLDE and find that, when the forcing term is small and we are in a regime where the solitary wave is stable, that the behavior of the solutions of the collective coordinate equations agrees very well with the numerical simulations. We found that the time evolution of the collective coordinates of the solitary wave in our numerical simulations, namely the position of the average charge density and the momentum of the solitary wave, provide good indicators for when the solitary wave first becomes unstable. When these variables stop being smooth functions of time (t), then the solitary wave starts to distort in shape.

Rectified motion of a Bose-Einstein condensate in a horizontally vibrating shallow optical lattice

We consider a Bose-Einstein condensate, described by the Gross-Pitaevskii equation, in a horizontally vibrating shallow optical lattice. We study the dynamics of a bright soliton using the collective coordinate approximation. We show that depending on the parameters, amplitude, and frequency of the vibration of the lattice, the phase space of the equation of motion for the soliton center of mass shows multistability. In the frequency locked regions, in which the soliton has a nonzero average velocity determined by the external frequency, the motion is quasiperiodic, and between the locked regions the soliton moves chaotically.

Gröbli solution for three magnetic vortices

The dynamics of N point vortices in a fluid is described by the Helmholtz–Kirchhoff (HK) equations which lead to a completely integrable Hamiltonian system for N=2 or 3 but chaotic dynamics for N&amp;gt;3. Here we consider a generalization of the HK equations to describe the dynamics of magnetic vortices within a collective-coordinate approximation. In particular, we analyze in detail the dynamics of a system of three magnetic vortices by a suitable generalization of the solution for three point vortices in an ordinary fluid obtained by Gröbli more than a century ago. The significance of our results for the dynamics of ferromagnetic elements is briefly discussed.

Macroscopic Quantum Tunneling of Two-Component Bose-Einstein Condensates

We show theoretically the existence of the metastable state and study the decay through the macroscopic quantum tunneling in two-component Bose-Einstein condensates. The numerical analysis of the coupled Gross-Pitaevskii equations clarifies the metastable states whose configuration preserves or breaks the symmetry of the trapping potential, depending on the interspecies interaction and the particle number. We calculate the tunneling decay rate of the metastable state by using the collective coordinate approximation and the bounce technique. It is found that the macroscopic quantum tunneling is observable in a wide range of the particle number.

Variational dynamics of Bose-Einsteincondensates in deep optical lattices

We study the dynamics of a Bose-Einstein condensatewavepacket trapped in an optical lattice. In the tight-bindinglimit, the system maps onto a discrete nonlinear Schrödingerequation. Using a collective coordinate approximation, weinvestigate different dynamical regimes: diffusive (in whichthe boson wavepacket spreads out), self-trapped (with aself-induced pinning), breathing (where the centre of mass ofthe wavepacket moves with an oscillating width), and solitonic(in which the wavepacket shape is exactly preserved duringthe dynamics).

Soliton scattering: the collective coordinate approximation

The model, with and without a generalized Hopf term, is studied using the collective coordinate approximation. In the spirit of this approximation, an ansatz is given for the head-on collision of two solitons. The equations of motion for the collective coordinates are then solved analytically, for solitons close together and for solitons far apart. The solutions show how the generalized Hopf term changes the scattering angle, which in its absence, is .

|ΔS|=1 hadronic weak decays of hyperons in a soliton model

Abstract We study the parity violating hyperon non-leptonic weak decays in the three flavor Skyrme model. We follow the approach in which the symmetry breaking terms in the action are diagonalized exactly within the collective coordinate approximation. We show that although this method introduces some configuration mixing, the ΔI =1/2 rule is numerically well satisfied. In addition, and in contrast to previous calculations, we show that not only the relative amplitudes are in good agreement with the empirical values but also their absolute values. The issue of whether the strong interaction enhancement factors should be included in soliton calculations is also addressed.

Classical and quantum kink scattering

We study the scattering of sine-Gordon kinks in both the classical and quantum case using a collective coordinate approximation. Since the sine-Gordon model is an exactly integrible theory this allows us to examine the validity of this approximation, which is used extensively in the study of solitons in both classical and quantum field theory.