Speed-of-light pulses in a massless nonlinear Dirac equation.

In this work, we explore a massless nonlinear Dirac equation, i.e., a nonlinear Weyl equation.

Speed-of-light pulses in the massless nonlinear Dirac equation with a potential.

We consider the massless nonlinear Dirac (NLD) equation in 1+1 dimension with scalar-scalar self-interaction g^{2}/2(Ψ[over ¯]Ψ)^{2} in the presence of three external electromagnetic real potentials V(x), a potential barrier, a constant potential, and a potential well. By solving numerically the NLD equation, we find different scenarios depending on initial conditions, namely, propagation of the initial pulse along one direction, splitting of the initial pulse into two pulses traveling in opposite directions, and focusing of two initial pulses followed by a splitting. For all considered cases, the final waves travel with the speed of light and are solutions of the massless linear Dirac equation.

Variational approach to studying solitary waves in the nonlinear Schrödinger equation with complex potentials.

We discuss the behavior of solitary wave solutions of the nonlinear Schrödinger equation (NLSE) as they interact with complex potentials, using a four-parameter variational approximation based on a dissipation functional formulation of the dynamics. We concentrate on spatially periodic potentials with the periods of the real and imaginary part being either the same or different. Our results for the time evolution of the collective coordinates of our variational ansatz are in good agreement with direct numerical simulation of the NLSE.

Erratum: Nonlinear Dirac equation solitary waves in external fields [Phys. Rev. E 86, 046602 (2012)].

This corrects the article DOI: 10.1103/PhysRevE.86.

Soliton dynamics in optical fibers using the generalized traveling-wave method.

The generalized traveling wave method (GTWM) is applied to the nonlinear Schrödinger (NLS) equation with general perturbations in order to obtain the equations of motion for an ansatz with six collective coordinates, namely the soliton position, the amplitude, the inverse of the soliton width, the velocity, the chirp, and the phase. The advantage of the new ansatz is that it yields three pairs of canonically conjugated coordinates and momenta that all are well-behaved. The new ansatz is applied to model the dynamics of a soliton in a dispersion-shifted optical fiber described by the generalized NLS, including dissipation, higher-order dispersion, Raman scattering, and self-steepening perturbations.

Soliton stability criterion for generalized nonlinear Schrödinger equations.

A stability criterion for solitons of the driven nonlinear Schrödinger equation (NLSE) has been conjectured. The criterion states that p'(v)<0 is a sufficient condition for instability, while p'(v)>0 is a necessary condition for stability; here, v is the soliton velocity and p=P/N, where P and N are the soliton momentum and norm, respectively. To date, the curve p(v) was calculated approximately by a collective coordinate theory, and the criterion was confirmed by simulations.

Stability of solitary waves in the nonlinear Dirac equation with arbitrary nonlinearity.

We consider the nonlinear Dirac equation in 1 + 1 dimension with scalar-scalar self interaction g(2)/κ+1(̅ΨΨ)(κ+1) and with mass m. Using the exact analytic form for rest frame solitary waves of the form Ψ(x,t)=ψ(x)e(-iωt) for arbitrary κ, we discuss the validity of various approaches to understanding stability that were successful for the nonlinear Schrödinger equation. In particular we study the validity of a version of Derrick's theorem and the criterion of Bogolubsky as well as the Vakhitov-Kolokolov criterion, and find that these criteria yield inconsistent results.

Propulsion efficiency of a dynamic self-assembled helical ribbon.

We study the dynamic self-assembly and propulsion of a ribbon formed from paramagnetic colloids in a dynamic magnetic field. The sedimented ribbon assembles due to time averaged dipolar interactions between the beads. The time dependence of the dipolar interactions together with hydrodynamic interactions cause a twisted ribbon conformation.

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.

Forced nonlinear Schrödinger equation with arbitrary nonlinearity.

We consider the nonlinear Schrödinger equation (NLSE) in 1+1 dimension with scalar-scalar self-interaction g(2)/κ+1(ψ*ψ)(κ+1) in the presence of the external forcing terms of the form re(-i(kx+θ))-δψ. We find new exact solutions for this problem and show that the solitary wave momentum is conserved in a moving frame where v(k)=2k. These new exact solutions reduce to the constant phase solutions of the unforced problem when r→0.

Refined empirical stability criterion for nonlinear Schrödinger solitons under spatiotemporal forcing.

We investigate the dynamics of traveling oscillating solitons of the cubic nonlinear Schrödinger (NLS) equation under an external spatiotemporal forcing of the form f(x,t)=aexp[iK(t)x]. For the case of time-independent forcing, a stability criterion for these solitons, which is based on a collective coordinate theory, was recently conjectured. We show that the proposed criterion has a limited applicability and present a refined criterion which is generally applicable, as confirmed by direct simulations.

Generalized traveling-wave method, variational approach, and modified conserved quantities for the perturbed nonlinear Schrödinger equation.

The generalized traveling wave method (GTWM) is developed for the nonlinear Schrödinger equation (NLSE) with general perturbations in order to obtain the equations of motion for an arbitrary number of collective coordinates. Regardless of the particular ansatz that is used, it is shown that this alternative approach is equivalent to the Lagrangian formalism, but has the advantage that only the Hamiltonian of the unperturbed system is required, instead of the Lagrangian for the perturbed system. As an explicit example, we take 4 collective coordinates, namely the position, velocity, amplitude and phase of the soliton, and show that the GTWM yields the same equations of motion as the perturbation theory based on the Inverse Scattering Transform and as the time variation of the norm, first moment of the norm, momentum, and energy for the perturbed NLSE.

Nonlinear Schrödinger equation with spatiotemporal perturbations.

We investigate the dynamics of solitons of the cubic nonlinear Schrödinger equation (NLSE) with the following perturbations: nonparametric spatiotemporal driving of the form f(x,t)=a exp[iK(t)x], damping, and a linear term which serves to stabilize the driven soliton. Using the time evolution of norm, momentum and energy, or, alternatively, a Lagrangian approach, we develop a collective-coordinate-theory which yields a set of ordinary differential equations (ODEs) for our four collective coordinates. These ODEs are solved analytically and numerically for the case of a constant, spatially periodic force f(x).

Driven and damped double sine-Gordon equation: the influence of internal modes on the soliton ratchet mobility.

This work studies the damped double sine-Gordon equation driven by a biharmonic force, where a parameter lambda controls the existence and the frequency of an internal mode. The role of internal oscillations of the kink width in ratchet dynamics of kink is investigated within the framework of collective coordinate theories. It is found that the ratchet velocity of the kink, when an internal mode appears in this system, decreases contrary to what was expected.

Sine-Gordon ratchets with general periodic, additive, and parametric driving forces.

We study the soliton ratchets in the damped sine-Gordon equation with periodic nonsinusoidal, additive, and parametric driving forces. By means of symmetry analysis of this system we show that the net motion of the kink is not possible if the frequencies of both forces satisfy a certain relationship. Using a collective coordinate theory with two degrees of freedom, we show that the ratchet motion of kinks appears as a consequence of a resonance between the oscillations of the momentum and the width of the kink.

Thermal diffusion of Boussinesq solitons.

We consider the problem of the soliton dynamics in the presence of an external noisy force for the Boussinesq type equations. A set of ordinary differential equations (ODEs) of the relevant coordinates of the system is derived. We show that for the improved Boussinesq (IBq) equation the set of ODEs has limiting cases leading to a set of ODEs which can be directly derived either from the ill-posed Boussinesq equation or from the Korteweg-de Vries (KdV) equation.

Vortex polarity switching by a spin-polarized current.

The spin-transfer effect is investigated for the vortex state of a magnetic nanodot. A spin current is shown to act similarly to an effective magnetic field perpendicular to the nanodot. Then a vortex with magnetization (polarity) parallel to the current polarization is energetically favorable.

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.

Inhomogeneous soliton ratchets under two ac forces.

We extend our previous work on soliton ratchet devices [L. Morales-Molina, Eur. Phys.

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.

Nondissipative diffusion of lattice solitons out of thermal equilibrium.

We perform Langevin dynamics simulations for pulse solitons on atomic chains with anharmonic nearest-neighbor interactions. After switching off noise and damping after a sufficiently long time, the solitons are only influenced by the thermal phonon bath which had been created by the noise. The soliton diffusion constant D is considerably smaller than before the switch-off, and it is proportional to the square of the temperature T , in contrast to the diffusion due to the noise which is proportional to T .

Ratchet behavior in nonlinear Klein-Gordon systems with pointlike inhomogeneities.

We investigate the ratchet dynamics of nonlinear Klein-Gordon kinks in a periodic, asymmetric lattice of pointlike inhomogeneities. We explain the underlying rectification mechanism within a collective coordinate framework, which shows that such a system behaves as a rocking ratchet for point particles. Careful attention is given to the kink width dynamics and its role in the transport.

Importance of the internal shape mode in magnetic vortex dynamics.

We investigate the motion of a nonplanar vortex in a circular easy-plane magnet with a rotating in-plane magnetic field. Our numerical simulations of the Landau-Lifshitz equations show that the vortex tends to a circular limit trajectory, with an orbit frequency which is lower than the driving field frequency. To describe this we develop a new collective variable theory by introducing additional variables which account for the internal degrees of freedom of the vortex core, strongly coupled to the translational motion.

Internal mode mechanism for collective energy transport in extended systems.

We study directed energy transport in homogeneous nonlinear extended systems in the presence of homogeneous ac forces and dissipation. We show that the mechanism responsible for unidirectional motion of topological excitations is the coupling of their internal and translation degrees of freedom. Our results lead to a selection rule for the existence of such motion based on resonances that explain earlier symmetry analysis of this phenomenon.

Rice's ansatz for overdamped phi4 kinks at finite temperature.

The dynamics of a kink driven by noise is analyzed using the two collective variables of the Rice ansatz: position and width. Starting from a stochastic partial differential equation, with the phi(4) potential in the overdamped limit, the pair of stochastic differential equations for the collective variables are derived without approximation other than the ansatz itself. From the steady state probability density of the kink width, the diffusivity of a kink is calculated.