arXiv subjects

P. G. Kevrekidis

Publications and source records attributed to P. G. Kevrekidis.

At least 451 records · Page 25Linked to original sources

arXiv2007

Symmetry Breaking in Linearly Coupled Dynamical Lattices

We examine one- and two-dimensional (1D and 2D) models of linearly coupled lattices of the discrete-nonlinear-Schr{ö}dinger type. Analyzing ground states of the systems with equal powers in the two components, we find a symmetry-breaking phenomenon beyond a critical value of the squared $l^2$-norm. Asymmetric states, with unequal powers in their components, emerge through a subcritical pitchfork bifurcation, which, for very weakly coupled lattices, changes into a supercritical one. We identify the stability of various solution branches. Dynamical manifestations of the symmetry breaking are studied by simulating the evolution of the unstable branches. The results present the first example of spontaneous symmetry breaking in 2D lattice solitons. This feature has no counterpart in the continuum limit, because of the collapse instability in the latter case.

nlin.PS↗

arXiv2007

Stability of Waves in Multi-component DNLS system

In this work, we systematically generalize the Evans function methodology to address vector systems of discrete equations. We physically motivate and mathematically use as our case example a vector form of the discrete nonlinear Schrodinger equation with both nonlinear and linear couplings between the components. The Evans function allows us to qualitatively predict the stability of the nonlinear waves under the relevant perturbations and to quantitatively examine the dependence of the corresponding point spectrum eigenvalues on the system parameters. These analytical predictions are subsequently corroborated by numerical computations.

nlin.PS↗

arXiv2007

Symmetry breaking bifurcation in Nonlinear Schrodinger /Gross-Pitaevskii Equations

We consider a class of nonlinear Schrodinger / Gross-Pitaveskii (NLS-GP) equations, i.e. NLS with a linear potential. We obtain conditions for a symmetry breaking bifurcation in a symmetric family of states as N, the squared L^2 norm (particle number, optical power), is increased. In the special case where the linear potential is a double-well with well separation L, we estimate N_{cr}, the symmetry breaking threshold. Along the ``lowest energy'' symmetric branch, there is an exchange of stability from the symmetric to asymmetric branch as N is increased beyond N_{cr}.

nlin.PS↗

arXiv2007

Modulated Amplitude Waves in Collisionally Inhomogeneous Bose-Einstein Condensates

We investigate the dynamics of an effectively one-dimensional Bose-Einstein condensate (BEC) with scattering length $a$ subjected to a spatially periodic modulation, $a=a(x)=a(x+L)$. This "collisionally inhomogeneous" BEC is described by a Gross-Pitaevskii (GP) equation whose nonlinearity coefficient is a periodic function of $x$. We transform this equation into a GP equation with constant coefficient $a$ and an additional effective potential and study a class of extended wave solutions of the transformed equation. For weak underlying inhomogeneity, the effective potential takes a form resembling a superlattice, and the amplitude dynamics of the solutions of the constant-coefficient GP equation obey a nonlinear generalization of the Ince equation. In the small-amplitude limit, we use averaging to construct analytical solutions for modulated amplitude waves (MAWs), whose stability we subsequently examine using both numerical simulations of the original GP equation and fixed-point computations with the MAWs as numerically exact solutions. We show that "on-site" solutions, whose maxima correspond to maxima of $a(x)$, are significantly more stable than their "off-site" counterparts.

nlin.PS↗

arXiv2007

From Feshbach-Resonance Managed Bose-Einstein Condensates to Anisotropic Universes: Some Applications of the Ermakov-Pinney equation with Time-Dependent Nonlinearity

In this work we revisit the topic of two-dimensional Bose-Einstein condensates under the influence of time-dependent magnetic confinement and time-dependent scattering length. A moment approach reduces the examination of moments of the wavefunction (in particular, of its width) to an Ermakov-Pinney (EP) ordinary differential equation (ODE). We use the well-known structure of the solutions of this nonlinear ODE to ``engineer'' trapping and interatomic interaction conditions that lead to condensates dispersing, breathing or even collapsing. The advantage of the approach is that it is fully tractable analytically, in excellent agreement with our numerical observations. As an aside, we also discuss how similar time-dependent EP equations may arise in the description of anisotropic scalar field cosmologies.

cond-mat.other↗

arXiv2007

Cerenkov-like radiation in a binary Schr{ö}dinger flow past an obstacle

We consider the dynamics of two coupled miscible Bose-Einstein condensates, when an obstacle is dragged through them. The existence of two different speeds of sound provides the possibility for three dynamical regimes: when both components are subcritical, we do not observe nucleation of coherent structures; when both components are supercritical they both form dark solitons in one dimension (1D) and vortices or rotating vortex dipoles in two dimensions (2D); in the intermediate regime, we observe the nucleation of a structure in the form of a dark-antidark soliton in 1D; subcritical component; the 2D analog of such a structure, a vortex-lump, is also observed.

cond-mat.soft↗

arXiv2007

Crossover dark soliton dynamics in ultracold one-dimensional Bose gases

Ultracold confined one-dimensional atomic gases are predicted to support dark soliton solutions arising from a nonlinear Schrödinger equation of suitable nonlinearity. In weakly-interacting (high density) gases, the nonlinearity is cubic, whereas an approximate model for describing the behaviour of strongly - interacting (low density) gases is one characterized by a quintic nonlinearity. We use an approximate analytical expression for the form of the nonlinearity in the intermediate regimes to show that, near the crossover between the two different regimes, the soliton is predicted and numerically confirmed to oscillate at a frequency of $\sqrt{2/3}Ω$, where $Ω$ is the harmonic trap frequency.

cond-mat.other↗

arXiv2006

Modulational instability in a layered Kerr medium: Theory and Experiment

We present the first experimental investigation of modulational instability in a layered Kerr medium. The particularly interesting and appealing feature of our configuration, consisting of alternating glass-air layers, is the piecewise-constant nature of the material properties, which allows a theoretical linear stability analysis leading to a Kronig-Penney equation whose forbidden bands correspond to the modulationally unstable regimes. We find very good {\it quantitative} agreement between theoretical, numerical, and experimental diagnostics of the modulational instability. Because of the periodicity in the evolution variable arising from the layered medium, there are multiple instability regions rather than just one as in the uniform medium.

nlin.PS↗

arXiv2006

Lyapunov--Schmidt reduction algorithm for three-dimensional discrete vortices

We address persistence and stability of three-dimensional vortex configurations in the discrete nonlinear Schrödinger (NLS) equation and develop a symbolic package based on Wolfram's MATHEMATICA for computations of the Lyapunov--Schmidt reduction method. The Lyapunov--Schmidt reduction method is a theoretical tool which enables us to study continuations and terminations of the discrete vortices for small coupling between lattice nodes as well as the spectral stability of the persistent configurations. The method was developed earlier in the context of the two-dimensional NLS lattice and applied to the on-site and off-site configurations (called the vortex cross and the vortex cell) by using semi-analytical computations. The present treatment develops a full symbolic computational package which takes a desired waveform at the anti-continuum limit of uncoupled sites, performs a required number of Lyapunov--Schmidt reductions and outputs the predictions on whether the configuration persists, for finite coupling, in the three-dimensional lattice and whether it is stable or unstable. It also provides approximations for the eigenvalues of the linearized stability problem. We report a number of applications of the algorithm to important multi-site configurations, such as the simple cube, the double cross, and the diamond. For each three-dimensional configuration, we identify exactly one solution, which is stable for small coupling between lattice nodes.

nlin.PS↗

arXiv2006

High-Order-Mode Soliton Structures in Two-Dimensional Lattices with Defocusing Nonlinearity

While fundamental-mode discrete solitons have been demonstrated with both self-focusing and defocusing nonlinearity, high-order-mode localized states in waveguide lattices have been studied thus far only for the self-focusing case. In this paper, the existence and stability regimes of dipole, quadrupole and vortex soliton structures in two-dimensional lattices induced with a defocusing nonlinearity are examined by the theoretical and numerical analysis of a generic envelope nonlinear lattice model. In particular, we find that the stability of such high-order-mode solitons is quite different from that with self-focusing nonlinearity. As a simple example, a dipole (``twisted'') mode soliton which may be stable in the focusing case becomes unstable in the defocusing regime. Our results may be relevant to other two-dimensional defocusing periodic nonlinear systems such as Bose-Einstein condensates with a positive scattering length trapped in optical lattices.

nlin.PS↗

arXiv2006

Fractional-Period Excitations in Continuum Periodic Systems

We investigate the generation of fractional-period states in continuum periodic systems. As an example, we consider a Bose-Einstein condensate confined in an optical-lattice potential. We show that when the potential is turned on non-adiabatically, the system explores a number of transient states whose periodicity is a fraction of that of the lattice. We illustrate the origin of fractional-period states analytically by treating them as resonant states of a parametrically forced Duffing oscillator and discuss their transient nature and potential observability.

cond-mat.other↗

arXiv2006

Dark Solitons in Discrete Lattices: Saturable versus Cubic Nonlinearities

In the present work, we study dark solitons in dynamical lattices with the saturable nonlinearity and compare them with those in lattices with the cubic nonlinearity. This comparison has become especially relevant in light of recent experimental developments in the former context. The stability properties of the fundamental waves, for both on-site and inter-site modes, are examined analytically and corroborated by numerical results. Furthermore, their dynamical evolution when they are found to be unstable is obtained through appropriately crafted numerical experiments.

nlin.PS↗

arXiv2006

Discrete Nonlinear Schrodinger Equations Free of the Peierls-Nabarro Potential

We derive a class of discrete nonlinear Schr{ö}dinger (DNLS) equations for general polynomial nonlinearity whose stationary solutions can be found from a reduced two-point algebraic problem. It is demonstrated that the derived class of discretizations contains subclasses conserving classical norm or a modified norm and classical momentum. These equations are interesting from the physical standpoint since they support stationary discrete solitons free of the Peierls-Nabarro potential. As a consequence, even in highly-discrete regimes, solitons are not trapped by the lattice and they can be accelerated by even weak external fields. Focusing on the cubic nonlinearity we then consider a small perturbation around stationary soliton solutions and, solving corresponding eigenvalue problem, we (i) demonstrate that solitons are stable; (ii) show that they have two additional zero-frequency modes responsible for their effective translational invariance; (iii) derive semi-analytical solutions for discrete solitons moving at slow speed. To highlight the unusual properties of solitons in the new discrete models we compare them with that of the classical DNLS equation giving several numerical examples.

nlin.PS↗

arXiv2006

Exact static solutions for discrete $ϕ^4$ models free of the Peierls-Nabarro barrier: Discretized first integral approach

We propose a generalization of the discrete Klein-Gordon models free of the Peierls-Nabarro barrier derived in Nonlinearity {\bf 12}, 1373 (1999) and Phys. Rev. E {\bf 72}, 035602(R) (2005), such that they support not only kinks but a one-parameter set of exact static solutions. These solutions can be obtained iteratively from a two-point nonlinear map whose role is played by the discretized first integral of the static Klein-Gordon field, as suggested in J. Phys. A {\bf 38}, 7617 (2005). We then discuss some discrete $ϕ^4$ models free of the Peierls-Nabarro barrier and identify for them the full space of available static solutions, including those derived recently in Phys. Rev. E {\bf 72} 036605 (2005) but not limited to them. These findings are also relevant to standing wave solutions of discrete nonlinear Schr{ö}dinger models. We also study stability of the obtained solutions. As an interesting aside, we derive the list of solutions to the continuum $ϕ^4$ equation that fill the entire two-dimensional space of parameters obtained as the continuum limit of the corresponding space of the discrete models.

nlin.PS↗

arXiv2006

Discrete surface solitons in two dimensions

We investigate fundamental localized modes in 2D lattices with an edge (surface). Interaction with the edge expands the stability area for ordinary solitons, and induces a difference between perpendicular and parallel dipoles; on the contrary, lattice vortices cannot exist too close to the border. Furthermore, we show analytically and numerically that the edge stabilizes a novel wave species, which is entirely unstable in the uniform lattice, namely, a "horseshoe" soliton, consisting of 3 sites. Unstable horseshoes transform themselves into a pair of ordinary solitons.

nlin.PS↗

arXiv2006

Radiationless Travelling Waves In Saturable Nonlinear Schrödinger Lattices

The longstanding problem of moving discrete solitary waves in nonlinear Schr{ö}dinger lattices is revisited. The context is photorefractive crystal lattices with saturable nonlinearity whose grand-canonical energy barrier vanishes for isolated coupling strength values. {\em Genuinely localised travelling waves} are computed as a function of the system parameters {\it for the first time}. The relevant solutions exist only for finite velocities.

nlin.PS↗

arXiv2006

Nonlinearity Management in Optics: Experiment, Theory, and Simulation

We conduct an experimental investigation of nonlinearity management in optics using femtosecond pulses and layered Kerr media consisting of glass and air. By examining the propagation properties over several diffraction lengths, we show that wave collapse can be prevented. We corroborate these experimental results with numerical simulations of the (2 + 1)-dimensional focusing cubic nonlinear Schrödinger equation with piecewise constant coefficients and a theoretical analysis of this setting using a moment method.

nlin.PS↗

arXiv2006

Mobility of Discrete Solitons in Quadratically Nonlinear Media

We study the mobility of solitons in second-harmonic-generating lattices. Contrary to what is known for their cubic counterparts, discrete quadratic solitons are mobile not only in the one-dimensional (1D) setting, but also in two dimensions (2D). We identify parametric regions where an initial kick applied to a soliton leads to three possible outcomes, namely, staying put, persistent motion, or destruction. For the 2D lattice, it is found that, for the solitary waves, the direction along which they can sustain the largest kick and can attain the largest speed is the diagonal. Basic dynamical properties of the discrete solitons are also discussed in the context of an analytical approximation, in terms of an effective Peierls-Nabarro potential in the lattice setting.

nlin.PS↗