arXiv subjects

Niurka R. Quintero

Publications and source records attributed to Niurka R. Quintero.

At least 37 records · Page 2Linked to original sources

arXiv2013

Nonlinear Dirac equation solitary waves in external fields

We consider the nonlinear Dirac equations (NLDE's) in 1+1 dimension with scalar-scalar self interaction $\frac{g^2}{κ+1} ({\bPsi} Ψ)^{κ+1}$ in the presence of various external electromagnetic fields. Starting from the exact solutions for the unforced problem 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 similar to the nonlinear Schr{ö}dinger equation (NLSE) that a sufficient dynamical condition for instability to arise is that $ dP(t)/d \dq(t) < 0$. Here $P(t)$ is the momentum of the solitary wave, and $\dq$ is the velocity of the center of the wave in the collective coordinate approximation. We found for our choices of external potentials we always have $ dP(t)/d \dq(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.

nlin.PS↗

arXiv2011

Forced Nonlinear Schroedinger Equation with Arbitrary Nonlinearity

We consider the nonlinear Schr{ö}dinger equation (NLSE) in 1+1 dimension with scalar-scalar self interaction $\frac{g^2}{κ+1} (ψ^\star ψ)^{κ+1}$ in the presence of the external forcing terms of the form $r e^{-i(kx + θ)} -δψ$. We find new exact solutions for this problem and show that the solitary wave momentum is conserved in a moving frame where $v_k=2 k$. These new exact solutions reduce to the constant phase solutions of the unforced problem when $r \rightarrow 0.$ In particular we study the behavior of solitary wave solutions in the presence of these external forces in a variational approximation which allows the position, momentum, width and phase of these waves to vary in time. We show that the stationary solutions of the variational equations include a solution close to the exact one and we study small oscillations around all the stationary solutions. We postulate that the dynamical condition for instability is that $ dp(t)/d \dot{q} (t) < 0$, where $p(t)$ is the normalized canonical momentum $p(t) = \frac{1}{M(t)} \frac {\partial L}{\partial {\dot q}}$, and $\dot{q}(t)$ is the solitary wave velocity. Here $M(t) = \int dx ψ^\star(x,t) ψ(x,t)$. Stability is also studied using a "phase portrait" of the soliton, where its dynamics is represented by two-dimensional projections of its trajectory in the four-dimensional space of collective coordinates. The criterion for stability of a soliton is that its trajectory is a closed single curve with a positive sense of rotation around a fixed point. We investigate the accuracy of our variational approximation and these criteria using numerical simulations of the NLSE.

nlin.PS↗

arXiv2011

A refined empirical stability criterion for nonlinear Schroedinger solitons under spatiotemporal forcing

We investigate the dynamics of travelling oscillating solitons of the cubic NLS equation under an external spatiotemporal forcing of the form $f(x,t) = a \exp[iK(t)x]$. 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. This includes more general situations where $K(t)$ is harmonic or biharmonic, with or without a damping term in the NLS equation. The refined criterion states that the soliton will be unstable if the "stability curve" $p(\v)$, where $p(t)$ and $\v(t)$ are the normalized momentum and the velocity of the soliton, has a section with a negative slope. Moreover, for the case of constant $K$ and zero damping we use the collective coordinate solutions to compute a "phase portrait" of the soliton where its dynamics is represented by two-dimensional projections of its trajectories in the four-dimensional space of collective coordinates. We conjecture, and confirm by simulations, that the soliton is unstable if a section of the resulting closed curve on the portrait has a negative sense of rotation.

nlin.PS↗

arXiv2011

Ratchet effect on a relativistic particle driven by external forces

We study the ratchet effect of a damped relativistic particle driven by both asymmetric temporal bi-harmonic and time-periodic piecewise constant forces. This system can be formally solved for any external force, providing the ratchet velocity as a non-linear functional of the driving force. This allows us to explicitly illustrate the functional Taylor expansion formalism recently proposed for this kind of systems. The Taylor expansion reveals particularly useful to obtain the shape of the current when the force is periodic, piecewise constant. We also illustrate the somewhat counterintuitive effect that introducing damping may induce a ratchet effect. When the force is symmetric under time-reversal and the system is undamped, under symmetry principles no ratchet effect is possible. In this situation increasing damping generates a ratchet current which, upon increasing the damping coefficient eventually reaches a maximum and decreases toward zero. We argue that this effect is not specific of this example and should appear in any ratchet system with tunable damping driven by a time-reversible external force.

nlin.PS↗

arXiv2010

Symmetries shape the current in ratchets induced by a bi-harmonic force

Equations describing the evolution of particles, solitons, or localized structures, driven by a zero-average, periodic, external force, and invariant under time reversal and a half-period time shift, exhibit a ratchet current when the driving force breaks these symmetries. The bi-harmonic force $f(t)=ε_1\cos(q ωt+ϕ_1)+ε_2\cos(pωt+ϕ_2)$ does it for almost any choice of $ϕ_{1}$ and $ϕ_{2}$, provided $p$ and $q$ are two co-prime integers such that $p+q$ is odd. It has been widely observed, in experiments in Josephson-junctions, photonic crystals, etc., as well as in simulations, that the ratchet current induced by this force has the shape $v\proptoε_1^pε_2^q\cos(p ϕ_{1} - q ϕ_{2} + θ_0)$ for small amplitudes, where $θ_0$ depends on the damping ($θ_0=π/2$ if there is no damping, and $θ_0=0$ for overdamped systems). We rigorously prove that this precise shape can be obtained solely from the broken symmetries of the system and is independent of the details of the equation describing the system.

nlin.PS↗

arXiv2009

Nonlinear Schrödinger Equation with Spatio-Temporal Perturbations

We investigate the dynamics of solitons of the cubic Nonlinear Schrödinger Equation (NLSE) with the following perturbations: non-parametric spatio-temporal driving of the form $f(x,t) = a \exp[i K(t) x]$, 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 ODEs for our four collective coordinates. These ODEs are solved analytically and numerically for the case of a constant, spatially periodic force $f(x)$. The soliton position exhibits oscillations around a mean trajectory with constant velocity. This means that the soliton performs, on the average, a unidirectional motion although the spatial average of the force vanishes. The amplitude of the oscillations is much smaller than the period of $f(x)$. In order to find out for which regions the above solutions are stable, we calculate the time evolution of the soliton momentum $P(t)$ and soliton velocity $V(t)$: This is a parameter representation of a curve $P(V)$ which is visited by the soliton while time evolves. Our conjecture is that the soliton becomes unstable, if this curve has a branch with negative slope. This conjecture is fully confirmed by our simulations for the perturbed NLSE. Moreover, this curve also yields a good estimate for the soliton lifetime: the soliton lives longer, the shorter the branch with negative slope is.

nlin.PS↗

arXiv2006

Does the dynamics of sine-Gordon solitons predict active regions of DNA?

In this work we analyze the possibility that soliton dynamics in a simple nonlinear model allows functionally relevant predictions of the behaviour of DNA. This suggestion was first put forward by Salerno [Phys. Rev. A, vol. 44, p. 5292 (1991)] by showing results indicating that sine-Gordon kinks were set in motion at certain regions of a DNA sequence that include promoters. We revisit that system and show that the observed behaviour has nothing to do with promoters; on the contrary, it originates from the bases at the boundary, which are not part of the studied genome. We explain this phenomenology in terms of an effective potential for the kink center. This is further extended to disprove recent claims that the dynamics of kinks [Lenholm and Hörnquist, Physica D, vol. 177, p. 233 (2003)] or breathers [Bashford, J. Biol. Phys., vol. 32, p. 27 (2006)] has functional significance. We conclude that no such information can be extracted from this simple nonlinear model or its associated effective potential.

q-bio.GN↗

arXiv2006

Reply to Comment on "Existence of Internal Modes of sine-Gordon Kinks"

In this reply to the comment by C. R. Willis, we show, by quoting his own statements, that the simulations reported in his original work with Boesch [Phys. Rev. B 42, 2290 (1990)] were done for kinks with nonzero initial velocity, in contrast to what Willis claims in his comment. We further show that his alleged proof, which assumes among other approximations that kinks are initially at rest, is not rigorous but an approximation. Moreover, there are other serious misconceptions which we discuss in our reply. As a consequence, our result that quasimodes do not exist in the sG equation [Phys. Rev. E 62, R60 (2000)] remains true.

cond-mat.stat-mech↗

arXiv2005

Soliton ratchets in homogeneous nonlinear Klein-Gordon systems

We study in detail the ratchet-like dynamics of topological solitons in homogeneous nonlinear Klein-Gordon systems driven by a bi-harmonic 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 necesary 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 $ϕ^4$ systems, which are seen to exhibit the same qualitative behavior. Our results are in agreement with recent experimental work on dissipation induced symmetry breaking.

cond-mat.supr-con↗

arXiv2005

On the ratchet effect

While the dependence of the directed transport on each of the ratchet-controlling parameters has been individually investigated experimentally, theoretically, and numerically, there is still no general criterion to apply to the whole set of these parameters to optimally control directed transport in general systems. We report that, to optimally enhance directed transport, there exists a universal force waveform which can be understood as the result of two competing fundamental mechanisms: the increase of the degree of breaking of symmetries and the decrease of the force impulse. We demonstrate that this universal waveform explains all the previous experimental, theoretical, and numerical results for a great diversity of systems, including motor enzymes.

physics.class-ph↗

arXiv2004

Comment on "Soliton ratchets induced by excitation of internal modes"

Very recently Willis et al. [Phys. Rev. E {\bf 69}, 056612 (2004)] have used a collective variable theory to explain the appearance of a nonzero energy current in an ac driven, damped sine-Gordon equation. In this comment, we prove rigorously that the time-averaged energy current in an ac driven nonlinear Klein-Gordon system is strictly zero.

nlin.SI↗

arXiv2004

Lagrangian Formalism in Perturbed Nonlinear Klein-Gordon Equations

We develop an alternative approach to study the effect of the generic perturbation (in addition to explicitly considering the loss term) in the nonlinear Klein-Gordon equations. By a change of the variables that cancel the dissipation term we are able to write the Lagrangian density and then, calculate the Lagrangian as a function of collective variables. We use the Lagrangian formalism together with the Rice {\it Ansatz} to derive the equations of motion of the collective coordinates (CCs) for the perturbed sine-Gordon (sG) and $ϕ^{4}$ systems. For the $N$ collective coordinates, regardless of the {\it Ansatz} used, we show that, for the nonlinear Klein-Gordon equations, this approach is equivalent to the {\it Generalized Traveling Wave Ansatz} ({\it GTWA})

nlin.SI↗

arXiv2004

Analytical approach to soliton ratchets in asymmetric potentials

We use soliton perturbation theory and collective coordinate ansatz to investigate the mechanism of soliton ratchets in a driven and damped asymmetric double sine-Gordon equation. We show that, at the second order of the perturbation scheme, the soliton internal vibrations can couple {\it effectively}, in presence of damping, to the motion of the center of mass, giving rise to transport. An analytical expression for the mean velocity of the soliton is derived. The results of our analysis confirm the internal mode mechanism of soliton ratchets proposed in [Phys. Rev. E {\bf 65} 025602(R) (2002)].

nlin.SI↗

arXiv2004

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.

cond-mat.stat-mech↗

arXiv2003

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 explains earlier symmetry analysis of this phenomenon. The direction of motion is found to depend both on the initial and the relative phases of the two harmonic drivings, even in the presence of noise.

cond-mat.stat-mech↗

arXiv2002

Using the Continuous Spectrum to "Feel" Integrability: the Effect of Boundary Conditions

The scope of this work is to propose a method for testing the integrability of a model partial differential (PDE) and/or differential difference equation (DDE). For monoparametric families of PDE/DDE's, that are known to possess isolated integrable points, we find that very special features occur in the continuous (``phonon'') spectrum at these ``singular'' points. We identify these features in the case example of a PDE and a DDE (that sustain front and pulse-like solutions respectively) for different types of boundary conditions. The key finding of the work is that such spectral features are generic near the singular, integrable points and hence we propose to explore a given PDE/DDE for such traits, as a means of assessing its potential integrability.

nlin.SI↗

arXiv2001

Anomalies of ac driven solitary waves with internal modes: Nonparametric resonances induced by parametric forces

We study the dynamics of kinks in the $ϕ^4$ model subjected to a parametric ac force, both with and without damping, as a paradigm of solitary waves with internal modes. By using a collective coordinate approach, we find that the parametric force has a non-parametric effect on the kink motion. Specifically, we find that the internal mode leads to a resonance for frequencies of the parametric driving close to its own frequency, in which case the energy of the system grows as well as the width of the kink. These predictions of the collective coordinate theory are verified by numerical simulations of the full partial differential equation. We finally compare this kind of resonance with that obtained for non-parametric ac forces and conclude that the effect of ac drivings on solitary waves with internal modes is exactly the opposite of their character in the partial differential equation.

cond-mat.stat-mech↗

arXiv2001

Soliton ratchets

The mechanism underlying the soliton ratchet, both in absence and in presence of noise, is investigated. We show the existence of an asymmetric internal mode on the soliton profile which couples, trough the damping in the system, to the soliton translational mode. Effective soliton transport is achieved when the internal mode and the external force are phase locked. We use as working model a generalized double sine-Gordon equation. The phenomenon is expected to be valid for generic soliton systems.

nlin.PS↗