arXiv subjects

P. G. Kevrekidis

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

At least 145 records · Page 8Linked to original sources

arXiv2019

Existence and spectral stability of multi-pulses in discrete Hamiltonian lattice systems

In the present work, we consider the existence and spectral stability of multi-pulse solutions in Hamiltonian lattice systems. We provide a general framework for the study of such wave patterns based on a discrete analogue of Lin's method, previously used in the continuum realm. We develop explicit conditions for the existence of $m$-pulse structures and subsequently develop a reduced matrix allowing us to address their spectral stability. As a prototypical example for the manifestation of the details of the formulation, we consider the discrete nonlinear Schrödinger equation. Different families of $2$- and $3$-pulse solitary waves are discussed, and analytical expressions for the corresponding stability eigenvalues are obtained which are in very good agreement with numerical results.

math.DS↗

arXiv2019

Extreme wave events for a nonlinear Schrödinger equation with linear damping and Gaussian driving

We perform a numerical study of the initial-boundary value problem, with vanishing boundary conditions, of a driven nonlinear Schrödinger equation (NLS) with linear damping and a Gaussian driver. We identify Peregrine-like rogue waveforms, excited by two different types of vanishing initial data decaying at an algebraic or exponential rate. The observed extreme events emerge on top of a decaying support. Depending on the spatial/temporal scales of the driver, the transient dynamics -- prior to the eventual decay of the solutions -- may resemble the one in the semiclassical limit of the integrable NLS, or may, e.g., lead to large-amplitude breather-like patterns. The effects of the damping strength and driving amplitude, in suppressing or enhancing respectively the relevant features, as well as of the phase of the driver in the construction of a diverse array of spatiotemporal patterns, are numerically analyzed.

nlin.PS↗

arXiv2019

Ring dark solitons in three-dimensional Bose-Einstein condensates

In this work we present a systematic study of the three-dimensional extension of the ring dark soliton examining its existence, stability, and dynamics in isotropic harmonically trapped Bose-Einstein condensates. Detuning the chemical potential from the linear limit, the ring dark soliton becomes unstable immediately, but can be fully stabilized by an external cylindrical potential. The ring has a large number of unstable modes which are analyzed through spectral stability analysis. Furthermore, a few typical destabilization dynamical scenarios are revealed with a number of interesting vortical structures emerging such as the two or four coaxial parallel vortex rings. In the process of considering the stability of the structure, we also develop a modified version of the degenerate perturbation theory method for characterizing the spectra of the coherent structure. This semi-analytical method can be reliably applied to any soliton with a linear limit to explore its spectral properties near this limit. The good agreement of the resulting spectrum is illustrated via a comparison with the full numerical Bogolyubov-de Gennes spectrum. The application of the method to the two-component ring dark-bright soliton is also discussed.

cond-mat.quant-gas↗

arXiv2019

Rogue Waves and Periodic Solutions of a Nonlocal Nonlinear Schrödinger Model

In the present work, a nonlocal nonlinear Schrödinger (NLS) model is studied by means of a recent technique that identifies solutions of partial differential equations, by considering them as fixed points in {\it space-time}. This methodology allows to perform a continuation of well-known solutions of the local NLS model to the nonlocal case. Four different examples of this type are presented, namely (a) the rogue wave in the form of the Peregrine soliton, (b) the generalization thereof in the form of the Kuznetsov-Ma breather, as well as two spatio-temporally periodic solutions in the form of elliptic functions. Importantly, all four waveforms can be continued in intervals of the parameter controlling the nonlocality of the model. The first two can be continued in a narrower interval, while the periodic ones can be extended to arbitrary nonlocalities and, in fact, present an intriguing bifurcation whereby they merge with (only) spatially periodic structures. The results suggest the generic relevance of rogue waves and related structures, as well as periodic solutions, in nonlocal NLS models.

nlin.PS↗

arXiv2019

Linear impurity modes in an electrical lattice: Theory and Experiment

We examine theoretically and experimentally the localized %and extended electrical modes existing in a bi-inductive electrical lattice containing a bulk or a surface capacitive impurity. By means of the formalism of lattice Green's functions, we are able to obtain closed-form expressions for the frequencies of the impurity (bound-state) eigenmodes and for their associated spatial profiles. This affords us a systematic understanding of how these mode properties change as a function of the system parameters. We test these analytical results against experimental measurements, in both the bulk and surface cases, and find very good agreement. Lastly, we turn to a series of quench experiments, where either a parameter of the lattice or the lattice geometry itself is rapidly switched between two values or configurations. In all cases, we are able to naturally explain the results of such quench experiments from the larger analytical picture that emerges as a result of the detailed characterization of the impurity-mode solution branches.

nlin.PS↗

arXiv2019

Reduced dynamics for one and two dark soliton stripes in the defocusing nonlinear Schrödinger equation: a variational approach

We study the dynamics and pairwise interactions of dark soliton stripes in the two-dimensional defocusing nonlinear Schrödinger equation. By employing a variational approach we reduce the dynamics for dark soliton stripes to a set of coupled one-dimensional "filament" equations of motion for the position and velocity of the stripe. The method yields good qualitative agreement with the numerical results as regards the transverse instability of the stripes. We propose a phenomenological amendment that also significantly improves the quantitative agreement of the method with the computations. Subsequently, the method is extended for a pair of symmetric dark soliton stripes that include the mutual interactions between the filaments. The reduced equations of motion are compared with a recently proposed adiabatic invariant method and its corresponding findings and are found to provide a more adequate representation of the original full dynamics for a wide range of cases encompassing perturbations with long and short wavelengths, and combinations thereof.

nlin.PS↗

arXiv2019

Dynamics of interacting dark soliton stripes

In the present work we examine the statics and dynamics of multiple parallel dark soliton stripes in a two-dimensional Bose-Einstein condensate. Our principal goal is to study the effect of the interaction between the stripes on the transverse instability of the individual stripes. We use a recently developed adiabatic invariant formulation to derive a quasi-analytical prediction for the stripe equilibrium position and for the Bogoliubov-de Gennes spectrum of excitations of stationary stripes. The cases of two-, three- and four-stripe states are studied in detail. We subsequently test our predictions against numerical simulations of the full two-dimensional Gross-Pitaevskii equation. We find that the number of unstable eigenmodes increases as the number of stripes increases due to (unstable) relative motions between the stripes. Their corresponding growth rates do not significantly change, although for large chemical potentials, the larger the stripe number, the larger the maximal instability growth rate. The instability induced dynamics of multiple stripe states and their decay into vortices are also investigated.

cond-mat.quant-gas↗

arXiv2019

Bright Solitary Waves on a Torus: Existence, Stability and Dynamics for the Nonlinear Schrödinger Model

Motivated by recent developments in the realm of matter waves, we explore the potential of creating solitary waves on the surface of a torus. This is an intriguing perspective due to the role of curvature in the shape and dynamics of the coherent structures. We find different families of bright solitary waves for attractive nonlinearities including ones localized in both angular directions, as well as waves localized in one direction and homogeneous in the other. The waves localized in both angular directions have also been partitioned into two types: those whose magnitude decays to zero and those who do not. The stability properties of the waves are examined and one family is found to be spectrally stable while most are spectrally unstable, a feature that we comment on. Finally, the nature of the ensuing nonlinear dynamics is touched upon.

nlin.PS↗

arXiv2019

Discrete breathers in $ϕ^4$ and related models

We touch upon the wide topic of discrete breather formation with a special emphasis on the the $ϕ^4$ model. We start by introducing the model and discussing some of the application areas/motivational aspects of exploring time periodic, spatially localized structures, such as the discrete breathers. Our main emphasis is on the existence, and especially on the stability features of such solutions. We explore their spectral stability numerically, as well as in special limits (such as the vicinity of the so-called anti-continuum limit of vanishing coupling) analytically. We also provide and explore a simple, yet powerful stability criterion involving the sign of the derivative of the energy vs. frequency dependence of such solutions. We then turn our attention to nonlinear stability, bringing forth the importance of a topological notion, namely the Krein signature. Furthermore, we briefly touch upon linearly and nonlinearly unstable dynamics of such states. Some special aspects/extensions of such structures are only touched upon, including moving breathers and dissipative variations of the model and some possibilities for future work are highlighted.

nlin.PS↗

arXiv2019

Two-dimensional rogue waves on zero background of the Davey-Stewartson II equation

A prototypical example of a rogue wave structure in a two-dimensional model is presented in the context of the Davey-Stewartson~II (DS~II) equation arising in water waves. The analytical methodology involves a Taylor expansion of an eigenfunctionof the model's Lax pair which is used to form a hierarchy of infinitely many new eigenfunctions. These are used for the construction of two-dimensional (2D) rogue waves (RWs) of the DS~II equation by the even-fold Darboux transformation (DT). The obtained 2D RWs, which are localized in both space and time, can be viewed as a 2D analogue of the Peregrine soliton and are thus natural candidates to describe oceanic RW phenomena,as well as ones in 2D fluid systems and water tanks.

nlin.SI↗

arXiv2019

Kink-kink and kink-antikink interactions with long-range tails

In this Letter, we address the {long-range interaction} between kinks and antikinks, as well as kinks and kinks, in $φ^{2n+4}$ field theories for $n>1$. The kink-antikink interaction is generically attractive, while the kink-kink interaction is generically repulsive. We find that the force of interaction decays with the $(\frac{2n}{n-1})$th power of their separation, and we identify the general prefactor for {\it arbitrary} $n$. Importantly, we test the resulting mathematical prediction with detailed numerical simulations of the dynamic field equation, and obtain good agreement between theory and numerics for the cases of $n=2$ ($φ^8$ model), $n=3$ ($φ^{10}$ model) and $n=4$ ($φ^{12}$ model).

hep-th↗

arXiv2019

Dynamics and stabilization of bright soliton stripes in the hyperbolic-dispersion nonlinear Schrödinger equation

We consider the dynamics and stability of bright soliton stripes in the two-dimensional nonlinear Schrödinger equation with hyperbolic dispersion, under the action of transverse perturbations. We start by discussing a recently proposed adiabatic-invariant approximation for transverse instabilities and its limitations in the bright soliton case. We then focus on a variational approximation used to reduce the dynamics of the bright-soliton stripe to effective equations of motion for its transverse shift. The reduction allows us to address the stripe's snaking instability, which is inherently present in the system, and follow the ensuing spatiotemporal undulation dynamics. Further, introducing a channel-shaped potential, we show that the instabilities (not only flexural, but also those of the necking type) can be attenuated, up to the point of complete stabilization of the soliton stripe.

nlin.PS↗

arXiv2019

Quasi-stable quantum vortex knots and links in anisotropic harmonically trapped Bose-Einstein condensates

Long-time existence of topologically nontrivial configurations of quantum vortices in the form of torus knots and links in trapped Bose-Einstein condensates is demonstrated numerically within the three-dimensional Gross-Pitaevskii equation with external anisotropic parabolic potential. We find out parametric domains near the trap anisotropy -- axial over planar frequency trapping ratio $λ\approx 1.5-1.6$ where the lifetime of such quasi-stationary rotating vortex structures is many hundreds of typical rotation times. This suggests the potential experimental observability of the structures. We quantify the relevant lifetimes as a function of the model parameters (e.g. $λ$) and initial condition parameters of the knot profile.

physics.atom-ph↗

arXiv2019

Rogue Waves as Self-Similar Solutions on a Background: A Direct Calculation

In the present work, we explore the possibility of developing rogue waves as exact solutions of some nonlinear dispersive equations, such as the nonlinear Schrödinger equation, but also, in a similar vein, the Hirota, Davey-Stewartson, and Zakharov models. The solutions that we find are ones previously identified through different methods. Nevertheless, they highlight an important aspect of these structures, namely their self-similarity. They thus offer an alternative tool in the very sparse (outside of the inverse scattering method) toolbox of attempting to identify analytically (or computationally) rogue wave solutions. This methodology is importantly independent of the notion of integrability. An additional nontrivial motivation for such a formulation is that it offers a frame in which the rogue waves are stationary. It is conceivable that in this frame one could perform a proper stability analysis of the structures.

nlin.PS↗

arXiv2019

Nonlinear Waves in an Experimentally Motivated Ring-shaped Bose-Einstein Condensate Setup

We systematically construct stationary soliton states in a one-component, two-dimensional, repulsive, Gross-Pitaevskii equation with a ring-shaped target-like trap similar to the potential used to confine a Bose-Einstein condensate in a recent experiment [Eckel, et al.\ {\em Nature} {\bf 506}, 200 (2014)]. In addition to the ground state configuration, we identify a wide variety of excited states involving phase jumps (and associated dark solitons) inside the ring. These configurations are obtained from a systematic bifurcation analysis starting from the linear, small atom density, limit. We study the stability, and when unstable, the dynamics of the most basic configurations. Often these lead to vortical dynamics inside the ring persisting over long time scales in our numerical experiments. To illustrate the relevance of the identified states, we showcase how such dark-soliton configurations (even the unstable ones) can be created in laboratory condensates by using phase-imprinting techniques.

cond-mat.quant-gas↗

arXiv2018

Solitary waves in the Ablowitz-Ladik equation with power-law nonlinearity

We introduce a generalized version of the Ablowitz-Ladik model with a power-law nonlinearity, as a discretization of the continuum nonlinear Schrödinger equation with the same type of the nonlinearity. The model opens a way to study the interplay of discreteness and nonlinearity features. We identify stationary discrete-soliton states for different values of nonlinearity power $σ$, and address changes of their stability as frequency $ω$ of the standing wave varies for given $σ$. Along with numerical methods, a variational approximation is used to predict the form of the discrete solitons, their stability changes, and bistability features by means of the Vakhitov-Kolokolov criterion (developed from the first principles). Development of instabilities and the resulting asymptotic dynamics are explored by means of direct simulations.

nlin.PS↗

arXiv2018

Solitary Waves of the Camassa-Holm Derivative Nonlinear Schrödinger equation

In this paper we examine a deformation of the derivative nonlinear Schrödinger (DNLS) equation, the so-called Camassa-Holm DNLS (CH-DNLS) equation. We use two asymptotic multiscale expansion methods to reduce this model to both the modified Korteweg-de Vries (MKdV) equation and the Korteweg-de Vries(KdV) equation. Using their exact soliton solutions, we construct approximate solutions of the original CH-DNLS equation, namely dark and anti-dark solitary waves. Direct numerical simulations demonstrate the validity of these approximate solutions and illustrate their dynamical evolution, including their potential for nearly elastic head-on collisions in the case examples considered.

nlin.PS↗

arXiv2018

Long-range interactions of kinks

We present a computational analysis of the long-range interactions of solitary waves in higher-order field theories. Our vehicle of choice is the $φ^8$ field theory, although we explore similar issues in example $φ^{10}$ and $φ^{12}$ models. In particular, we discuss the fundamental differences between the latter higher-order models and the standard $φ^4$ model. Upon establishing the power-law asymptotics of the model's solutions' approach towards one of the steady states, we make the case that such asymptotics require particular care in setting up multi-soliton initial conditions. A naive implementation of additive or multiplicative ansätze gives rise to highly pronounced radiation effects and eventually leads to the illusion of a repulsive interaction between a kink and an antikink in such higher-order field theories. We propose and compare several methods for how to "distill" the initial data into suitable ansätze, and we show how these approaches capture the attractive nature of interactions between the topological solitons in the presence of power-law tails (long-range interactions). This development paves the way for a systematic examination of solitary wave interactions in higher-order field theories and raises some intriguing questions regarding potential experimental observations of such interactions. As an Appendix, we present an analysis of kink-antikink interactions in the example models via the method of collective coordinates.

hep-th↗