arXiv subjects

R. Carretero-González

Publications and source records attributed to R. Carretero-González.

36 records · Page 2Linked to original sources

arXiv2011

Variational approximations in discrete nonlinear Schrödinger equations with next-nearest-neighbor couplings

Solitons of a discrete nonlinear Schrödinger equation which includes the next-nearest-neighbor interactions are studied by means of a variational approximation and numerical computations. A large family of multi-humped solutions, including those with a nontrivial phase structure which are a feature particular to the next-nearest-neighbor interaction model, are accurately predicted by the variational approximation. Bifurcations linking solutions with the trivial and nontrivial phase structures are also captured remarkably well, including a prediction of critical parameter values.

nlin.PS↗

arXiv2008

Azimuthal Modulational Instability of Vortices in the Nonlinear Schrödinger Equation

We study the azimuthal modulational instability of vortices with different topological charges, in the focusing two-dimensional nonlinear Schr{ö}dinger (NLS) equation. The method of studying the stability relies on freezing the radial direction in the Lagrangian functional of the NLS in order to form a quasi-one-dimensional azimuthal equation of motion, and then applying a stability analysis in Fourier space of the azimuthal modes. We formulate predictions of growth rates of individual modes and find that vortices are unstable below a critical azimuthal wave number. Steady state vortex solutions are found by first using a variational approach to obtain an asymptotic analytical ansatz, and then using it as an initial condition to a numerical optimization routine. The stability analysis predictions are corroborated by direct numerical simulations of the NLS. We briefly show how to extend the method to encompass nonlocal nonlinearities that tend to stabilize solutions.

nlin.PS↗

arXiv2008

Spinor Bose-Einstein condensate flow past an obstacle

We study the flow of a spinor (F=1) Bose-Einstein condensate in the presence of an obstacle. We consider the cases of ferromagnetic and polar spin-dependent interactions and find that the system demonstrates two speeds of sound that are identified analytically. Numerical simulations reveal the nucleation of macroscopic nonlinear structures, such as dark solitons and vortex-antivortex pairs, as well as vortex rings in one- and higher-dimensional settings respectively, when a localized defect (e.g., a blue-detuned laser beam) is dragged through the spinor condensate at a speed larger than the second critical speed.

nlin.PS↗

arXiv2007

Solitons in one-dimensional nonlinear Schrödinger lattices with a local inhomogeneity

In this paper we analyze the existence, stability, dynamical formation and mobility properties of localized solutions in a one-dimensional system described by the discrete nonlinear Schrödinger equation with a linear point defect. We consider both attractive and repulsive defects in a focusing lattice. Among our main findings are: a) the destabilization of the on--site mode centered at the defect in the repulsive case; b) the disappearance of localized modes in the vicinity of the defect due to saddle-node bifurcations for sufficiently strong defects of either type; c) the decrease of the amplitude formation threshold for attractive and its increase for repulsive defects; and d) the detailed elucidation as a function of initial speed and defect strength of the different regimes (trapping, trapping and reflection, pure reflection and pure transmission) of interaction of a moving localized mode with the defect.

nlin.PS↗

arXiv2005

Dynamics and Manipulation of Matter-Wave Solitons in Optical Superlattices

We analyze the existence and stability of bright, dark, and gap matter-wave solitons in optical superlattices. Then, using these properties, we show that (time-dependent) ``dynamical superlattices'' can be used to controllably place, guide, and manipulate these solitons. In particular, we use numerical experiments to displace solitons by turning on a secondary lattice structure, transfer solitons from one location to another by shifting one superlattice substructure relative to the other, and implement solitonic ``path-following'', in which a matter wave follows the time-dependent lattice substructure into oscillatory motion.

cond-mat.other↗

arXiv2005

Stability of dark solitons in a Bose-Einstein condensate trapped in an optical lattice

We investigate the stability of dark solitons (DSs) in an effectively one-dimensional Bose-Einstein condensate in the presence of the magnetic parabolic trap and an optical lattice (OL). The analysis is based on both the full Gross-Pitaevskii equation and its tight-binding approximation counterpart (discrete nonlinear Schr{ö}dinger equation). We find that DSs are subject to weak instabilities with an onset of instability mainly governed by the period and amplitude of the OL. The instability, if present, sets in at large times and it is characterized by quasi-periodic oscillations of the DS about the minimum of the parabolic trap.

cond-mat.other↗

arXiv2004

Nonlinear Lattice Dynamics of Bose-Einstein Condensates

The Fermi-Pasta-Ulam (FPU) model, which was proposed 50 years ago to examine thermalization in non-metallic solids and develop ``experimental'' techniques for studying nonlinear problems, continues to yield a wealth of results in the theory and applications of nonlinear Hamiltonian systems with many degrees of freedom. Inspired by the studies of this seminal model, solitary-wave dynamics in lattice dynamical systems have proven vitally important in a diverse range of physical problems--including energy relaxation in solids, denaturation of the DNA double strand, self-trapping of light in arrays of optical waveguides, and Bose-Einstein condensates (BECs) in optical lattices. BECS, in particular, due to their widely ranging and easily manipulated dynamical apparatuses--with one to three spatial dimensions, positive-to-negative tuning of the nonlinearity, one to multiple components, and numerous experimentally accessible external trapping potentials--provide one of the most fertile grounds for the analysis of solitary waves and their interactions. In this paper, we review recent research on BECs in the presence of deep periodic potentials, which can be reduced to nonlinear chains in appropriate circumstances. These reductions, in turn, exhibit many of the remarkable nonlinear structures (including solitons, intrinsic localized modes, and vortices) that lie at the heart of the nonlinear science research seeded by the FPU paradigm.

nlin.PS↗

arXiv2004

Vortices in a Bose-Einstein condensate confined by an optical lattice

We investigate the dynamics of vortices in repulsive Bose-Einstein condensates in the presence of an optical lattice (OL) and a parabolic magnetic trap. The dynamics is sensitive to the phase of the OL potential relative to the magnetic trap, and depends less on the OL strength. For the cosinusoidal OL potential, a local minimum is generated at the trap's center, creating a stable equilibrium for the vortex, while in the case of the sinusoidal potential, the vortex is expelled from the center, demonstrating spiral motion. Cases where the vortex is created far from the trap's center are also studied, revealing slow outward-spiraling drift. Numerical results are explained in an analytical form by means of a variational approximation. Finally, motivated by a discrete model (which is tantamount to the case of the strong OL lattice), we present a novel type of vortex consisting of two pairs of anti-phase solitons.

cond-mat.other↗

arXiv2004

Three-Dimensional Solitary Waves and Vortices in a Discrete Nonlinear Schr{ö}dinger Lattice

In a benchmark dynamical-lattice model in three dimensions, the discrete nonlinear Schr{ö}dinger equation, we find discrete vortex solitons with various values of the topological charge $S$. Stability regions for the vortices with $S=0,1,3$ are investigated. The S=2 vortex is unstable, spontaneously rearranging into a stable one with S=3. In a two-component extension of the model, we find a novel class of stable structures, consisting of vortices in the different components, perpendicularly oriented to each other. Self-localized states of the proposed types can be observed experimentally in Bose-Einstein condensates trapped in optical lattices, and in photonic crystals built of microresonators.

cond-mat.other↗

arXiv2004

Domain Walls of Single-Component Bose-Einstein Condensates in External Potentials

We demonstrate the possibility of creating domain walls described by a single component Gross-Pitaevskii equation with attractive interaction, in the presence of an optical-lattice potential. While it is found that the extended domain wall is unstable, we show that the external magnetic trap can stabilize it. Stable solutions include twisted domain walls, as well as asymmetric solitons. The results also apply to spatial solitons in planar waveguides with transverse modulation of the refractive index.

cond-mat.other↗

arXiv2003

Families of Matter-Waves for Two-Component Bose-Einstein Condensates

We produce several families of solutions for two-component nonlinear Schrödinger/Gross-Pitaevskii equations. These include domain walls and the first example of an antidark or gray soliton in the one component, bound to a bright or dark soliton in the other. Most of these solutions are linearly stable in their entire domain of existence. Some of them are relevant to nonlinear optics, and all to Bose-Einstein condensates (BECs). In the latter context, we demonstrate robustness of the structures in the presence of parabolic and periodic potentials (corresponding, respectively, to the magnetic trap and optical lattices in BECs).

cond-mat↗

arXiv2002

Localized breathing solutions for Bose-Einstein condensates in periodic traps

We demonstrate the existence of localized oscillatory breathers for quasi-one-dimensional Bose-Einstein condensates confined in periodic potentials. The breathing behavior corresponds to position-oscillations of individual condensates about the minima of the potential lattice. We deduce the structural stability of the localized oscillations from the construction. The stability is confirmed numerically for perturbations to the initial state of the condensate, to the potential trap, as well as for external noise. We also construct periodic and chaotic extended oscillations for the chain of condensates. All our findings are verified by direct numerical integration of the Gross-Pitaevskii equation in one dimension.

cond-mat↗

arXiv2000

A quasi-diagonal approach to the estimation of Lyapunov spectra for spatio-temporal systems from multivariate time series

We describe methods of estimating the entire Lyapunov spectrum of a spatially extended system from multivariate time-series observations. Provided that the coupling in the system is short range, the Jacobian has a banded structure and can be estimated using spatially localised reconstructions in low embedding dimensions. This circumvents the ``curse of dimensionality'' that prevents the accurate reconstruction of high-dimensional dynamics from observed time series. The technique is illustrated using coupled map lattices as prototype models for spatio-temporal chaos and is found to work even when the coupling is not strictly local but only exponentially decaying.

nlin.CD↗

arXiv1999

Thermodynamic limit from small lattices of coupled maps

We compare the behaviour of a small truncated coupled map lattice with random inputs at the boundaries with that of a large deterministic lattice essentially at the thermodynamic limit. We find exponential convergence for the probability density, predictability, power spectrum, and two-point correlation with increasing truncated lattice size. This suggests that spatio-temporal embedding techniques using local observations cannot detect the presence of spatial extent in such systems and hence they may equally well be modelled by a local low dimensional stochastically driven system.

chao-dyn↗

arXiv1999

One-dimensional dynamics for travelling fronts in coupled map lattices

Multistable coupled map lattices typically support travelling fronts, separating two adjacent stable phases. We show how the existence of an invariant function describing the front profile, allows a reduction of the infinitely-dimensional dynamics to a one-dimensional circle homeomorphism, whose rotation number gives the propagation velocity. The mode-locking of the velocity with respect to the system parameters then typically follows. We study the behaviour of fronts near the boundary of parametric stability, and we explain how the mode-locking tends to disappear as we approach the continuum limit of an infinite density of sites.

chao-dyn↗

arXiv1998

Scaling and interleaving of sub-system Lyapunov exponents for spatio-temporal systems

The computation of the entire Lyapunov spectrum for extended dynamical systems is a very time consuming task. If the system is in a chaotic spatio-temporal regime it is possible to approximately reconstruct the Lyapunov spectrum from the spectrum of a sub-system in a very cost effective way. In this work we present a new rescaling method, which gives a significantly better fit to the original Lyapunov spectrum. It is inspired by the stability analysis of the homogeneous evolution in a one-dimensional coupled map lattice but appears to be equally valid in a much wider range of cases. We evaluate the performance of our rescaling method by comparing it to the conventional rescaling (dividing by the relative sub-system volume) for one and two-dimensional lattices in spatio-temporal chaotic regimes. In doing so we notice that the Lyapunov spectra for consecutive sub-system sizes are interleaved and we discuss the possible ways in which this may arise. Finally, we use the new rescaling to approximate quantities derived from the Lyapunov spectrum (largest Lyapunov exponent, Lyapunov dimension and Kolmogorov-Sinai entropy) finding better convergence as the sub-system size is increased than with conventional rescaling.

chao-dyn↗

arXiv1998

Low dimensional travelling interfaces in coupled map lattices

We study the dynamics of the travelling interface arising from a bistable piece-wise linear one-way coupled map lattice. We show how the dynamics of the interfacial sites, separating the two superstable phases of the local map, is finite dimensional and equivalent to a toral map. The velocity of the travelling interface corresponds to the rotation vector of the toral map. As a consequence, a rational velocity of the travelling interface is subject to mode-locking with respect to the system parameters. We analytically compute the Arnold's tongues where particular spatio-temporal periodic orbits exist. The boundaries of the mode-locked regions correspond to border-collision bifurcations of the toral map. By varying the system parameters it is possible to increase the number of interfacial sites corresponding to a border-collision bifurcation of the interfacial attracting cycle. We finally give some generalizations towards smooth coupled map lattices whose interface dynamics is typically infinite dimensional.

chao-dyn↗

arXiv1998

Mode-locking in coupled map lattices

We study propagation of pulses along one-way coupled map lattices, which originate from the transition between two superstable states of the local map. The velocity of the pulses exhibits a staircase-like behaviour as the coupling parameter is varied. For a piece-wise linear local map, we prove that the velocity of the wave has a Devil's staircase dependence on the coupling parameter. A wave travelling with rational velocity is found to be stable to parametric perturbations in a manner akin to rational mode-locking for circle maps. We provide evidence that mode-locking is also present for a broader range of maps and couplings.

chao-dyn↗