arXiv subjects

Renato Alvarez-Nodarse

Publications and source records attributed to Renato Alvarez-Nodarse.

9 recordsLinked to original sources

arXiv2026

A Note on the Paper "Localization of Zeros of Polar Polynomials on the Unit Disk"

We show that none of the principal claims of Costas-Santos and Rehouma, Rocky Mountain J. Math. 54 (2024), 995-1004, survives as a new and valid result. The main localisation theorem is false as stated; under the missing hypothesis of orthogonality on the unit circle, it is a classical exercise recorded by Marden and by Borwein and Erdélyi, while the published proof omits its decisive finite-degree estimate. The annular claim and Sendov application are unproved, and some reported numerical values contradict the latter under its required hypothesis. We establish instead a measure-independent Grace--Szego theorem with sharp affine and optimal disc bounds, and characterise exactly when the convolution method extends to a general factor. This provides the necessary correction to the published record.

math.CA↗

arXiv2026

Generalized PT-symmetric nonlinear Dirac equation: exact solitary waves solutions, stability and conservation laws

We derive an exact solitary wave solution for the $\PTb$-symmetric nonlinear Dirac equation with a scalar-scalar interaction. We consider a power-law nonlinearity of the form $|\barΨ\,Ψ|^{k}\,Ψ$ for positive values of $k$. The system's energy is conserved despite the presence of a gain-loss term, which is quantified by the parameter $Λ$. We show that the $\PTb$-transition point is defined by the solution's existence condition and is independent of the nonlinearity exponent $k$. Furthermore, momentum is conserved, although neither the canonical momentum nor the charge is a conserved quantity. A notable result is that the stationary solution, obtained from the continuity equations, exhibits nonzero momentum in its rest frame. We also derive a moving soliton solution, where the gain-loss parameter allows the soliton's velocity to be precisely chosen so that the moving soliton possesses zero momentum. Finally, we establish that the presence of a gain-loss mechanism and higher-order nonlinearity restrict the stability domain of the solutions.

nlin.PS↗

arXiv2023

The Schrödinger equation for the Rosen-Morse type potential revisited with applications

We rigorously solve the time-independent Schrödinger equation for the Rosen-Morse type potential. By using the Nikiforov-Uvarov method, we obtain, in a systematic way, the complete solution of such equation, which includes the so-called bound states (square-integrable solutions) associated with the discrete spectrum, as well as unbound states region (bounded but not necessarily square-integrable solutions) related to the continuous part of the spectrum. The resolution of this problem is used to show that the kinks of the non-linear Klein-Gordon equation with $φ^{2p+2}$ type potentials are stable. We also derive the orthogonality and completeness relations satisfied by the set of eigenfunctions which are useful in the description of the dynamics of kinks under perturbations or interacting with antikinks.

quant-ph↗

arXiv2022

Stability of solitary waves in nonlinear Klein-Gordon equations

The stability of topological solitary waves and pulses in one-dimensional nonlinear Klein-Gordon systems is revisited. The linearized equation describing small deviations around the static solution leads to a Sturm-Liouville problem, which is solved in a systematic way for the $-l\,(l+1)\,\sech^2(x)$-potential, showing the orthogonality and completeness relations fulfilled by the set of its solutions for all values $l\in\mathbb{N}$. This approach allows to determine the linear stability of kinks and pulses of certain nonlinear Klein-Gordon equations. Two families of novel nonlinear Klein-Gordon potentials are introduced. The exact solutions (kinks and pulses) for these potentials are exactly calculated, even when the nonlinear potential is not explicitly known. The kinks of the novel models are found to be stable, whereas the pulses are unstable. The stability of the pulses is achieved by introducing certain spatial inhomogeneities.

nlin.PS↗

arXiv2015

General approach for dealing with dynamical systems with spatiotemporal periodicities

Dynamical systems often contain oscillatory forces or depend on periodic potentials. Time or space periodicity is reflected in the properties of these systems through a dependence on the parameters of their periodic terms. In this paper we provide a general theoretical framework for dealing with these kinds of systems, regardless of whether they are classical or quantum, stochastic or deterministic, dissipative or nondissipative, linear or nonlinear, etc. In particular, we are able to show that simple symmetry considerations determine, to a large extent, how their properties depend functionally on some of the parameters of the periodic terms. For the sake of illustration, we apply this formalism to find the functional dependence of the expectation value of the momentum of a Bose-Einstein condensate, described by the Gross-Pitaewskii equation, when it is exposed to a sawtooth potential whose amplitude is periodically modulated in time. We show that, by using this formalism, a small set of measurements is enough to obtain the functional form for a wide range of parameters. This can be very helpful when characterizing experimentally the response of systems for which performing measurements is costly or difficult.

cond-mat.stat-mech↗

arXiv2014

Time-shift invariance determines the functional shape of the current in dissipative rocking ratchets

Ratchets are devices able to rectify an otherwise oscillatory behavior by exploiting an asymmetry of the system. In rocking ratchets the asymmetry is induced through a proper choice of external forces and modulations of nonlinear symmetric potentials. The ratchet currents thus obtained in systems as different as semiconductors, Josephson junctions, optical lattices, or ferrofluids, show a set of universal features. A satisfactory explanation for them has challenged theorist for decades, and so far we still lack a general theory of this phenomenon. Here we provide such a theory by exploring ---through functional analysis--- the constraints that the simple assumption of time-shift invariance of the ratchet current imposes on its dependence on the external drivings. Because the derivation is based on so general a principle, the resulting expression is valid irrespective of the details and the nature of the physical systems to which it is applied, and of whether they are classical, quantum, or stochastic. The theory also explains deviations observed from universality under special conditions, and allows to make predictions of phenomena not yet observed in any experiment or simulation.

nlin.PS↗

arXiv2013

Comment on "Ratchet universality in the presence of thermal noise"

A recent paper [Phys. Rev. E 87, 062114 (2013)] presents numerical simulations on a system exhibiting directed ratchet transport of a driven overdamped Brownian particle subjected to a spatially periodic, symmetric potential. The authors claim that their simulations prove the existence of a universal waveform of the external force which optimally enhances directed transport, hence confirming the validity of a previous conjecture put forward by one of them in the limit of vanishing noise intensity. With minor corrections due to noise, the conjecture holds even in the presence of noise, according to the authors. On the basis of their results the authors claim that all previous theories, which predict a different optimal force waveform, are incorrect. In this comment we provide sufficient numerical evidence showing that there is no such universal force waveform and that the evidence obtained by the authors otherwise is due to a fortunate choice of the parameters. Our simulations also suggest that previous theories correctly predict the shape of the optimal waveform within their validity regime, namely when the forcing is weak. On the contrary, the aforementioned conjecture is shown to be wrong.

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↗