arXiv subjects

P. G. Kevrekidis

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

At least 217 records · Page 12Linked to original sources

arXiv2017

Solitary waves in a two-dimensional nonlinear Dirac equation: from discrete to continuum

In the present work, we explore a nonlinear Dirac equation motivated as the continuum limit of a binary waveguide array model. We approach the problem both from a near-continuum perspective as well as from a highly discrete one. Starting from the former, we see that the continuum Dirac solitons can be continued for all values of the discretization (coupling) parameter, down to the uncoupled (so-called anti-continuum) limit where they result in a 9-site configuration. We also consider configurations with 1- or 2-sites at the anti-continuum limit and continue them to large couplings, finding that they also persist. For all the obtained solutions, we examine not only the existence, but also the spectral stability through a linearization analysis and finally consider prototypical examples of the dynamics for a selected number of cases for which the solutions are found to be unstable.

nlin.PS↗

arXiv2017

Dark Soliton Scattering in Symmetric and Asymmetric Double Potential Barriers

Motivated by the recent theoretical study of (bright) soliton diode effects in systems with multiple scatterers, as well as by experimental investigations of soliton-impurity interactions, we consider some prototypical case examples of interactions of dark solitons with a pair of scatterers. In a way fundamentally opposite to the case of bright solitons (but consonant to their "anti-particle character"), we find that dark solitons accelerate as they pass the first barrier and hence cannot be trapped by a second equal-height barrier. A pair of unequal barriers may lead to reflection from the second one, however trapping in the inter-barrier region cannot occur. We also give some examples of dynamical adjusting the barriers to trap the dark soliton in the inter-barrier region, yet we show that this can only occur over finite time horizons, with the dark soliton always escaping eventually, contrary again to what is potentially the case with bright solitons.

nlin.PS↗

arXiv2017

Revisiting Diffusion: Self-similar Solutions and the $t^{-1/2}$ Decay in Initial and Initial-Boundary Value Problems

The diffusion equation is a universal and standard textbook model for partial differential equations (PDEs). In this work, we revisit its solutions, seeking, in particular, self-similar profiles. This problem connects to the classical theory of special functions and, more specifically, to the Hermite as well as the Kummer hypergeometric functions. Reconstructing the solution of the original diffusion model from novel self-similar solutions of the associated self-similar PDE, we infer that the $t^{-1/2}$ decay law of the diffusion amplitude is {\it not necessary}. In particular, it is possible to engineer setups of {\it both} the Cauchy problem and the initial-boundary value problem in which the solution decays at a {\it different rate}. Nevertheless, we observe that the $t^{-1/2}$ rate corresponds to the dominant decay mode among integrable initial data, i.e., ones corresponding to finite mass. Hence, unless the projection to such a mode is eliminated, generically this decay will be the slowest one observed. In initial-boundary value problems, an additional issue that arises is whether the boundary data are \textit{consonant} with the initial data; namely, whether the boundary data agree at all times with the solution of the Cauchy problem associated with the same initial data, when this solution is evaluated at the boundary of the domain. In that case, the power law dictated by the solution of the Cauchy problem will be selected. On the other hand, in the non-consonant cases a decomposition of the problem into a self-similar and a non-self-similar one is seen to be beneficial in obtaining a systematic understanding of the resulting solution.

math.AP↗

arXiv2017

Single and Multiple Vortex Rings in Three-Dimensional Bose-Einstein Condensates: Existence, Stability and Dynamics

In the present work, we explore the existence, stability and dynamics of single and multiple vortex ring states that can arise in Bose-Einstein condensates. Earlier works have illustrated the bifurcation of such states, in the vicinity of the linear limit, for isotropic or anisotropic three-dimensional harmonic traps. Here, we extend these states to the regime of large chemical potentials, the so-called Thomas-Fermi limit, and explore their properties such as equilibrium radii and inter-ring distance, for multi-ring states, as well as their vibrational spectra and possible instabilities. In this limit, both the existence and stability characteristics can be partially traced to a particle picture that considers the rings as individual particles oscillating within the trap and interacting pairwise with one another. Finally, we examine some representative instability scenarios of the multi-ring dynamics including breakup and reconnections, as well as the transient formation of vortex lines.

nlin.PS↗

arXiv2017

Linear and nonlinear dynamics of isospectral granular chains

We study the dynamics of isospectral granular chains that are highly tunable due to the nonlinear Hertz contact law interaction between the granular particles. The system dynamics can thus be tuned easily from being linear to strongly nonlinear by adjusting the initial compression applied to the chain. In particular, we introduce both discrete and continuous spectral transformation schemes to generate a family of granular chains that are isospectral in their linear limit. Inspired by the principle of supersymmetry in quantum systems, we also introduce a methodology to add or remove certain eigenfrequencies, and we demonstrate numerically that the corresponding physical system can be constructed in the setting of one-dimensional granular crystals. In the linear regime, we highlight the commonalities and differences in the elastic wave transmission characteristics of such isospectral systems, and emphasize that the presented mathematical framework allows one to suitably tailor the wave transmission through a general class of granular chains. Moreover, we show how the dynamic response of these structures deviates from its linear limit as we introduce Hertzian nonlinearity in the chain and how nonlinearity breaks the notion of linear isospectrality.

nlin.PS↗

arXiv2017

Computing stationary solutions of the two-dimensional Gross-Pitaevskii equation with Deflated continuation

In this work we employ a recently proposed bifurcation analysis technique, the deflated continuation algorithm, to compute steady-state solitary waveforms in a one-component, two dimensional nonlinear Schrödinger equation with a parabolic trap and repulsive interactions. Despite the fact that this system has been studied extensively, we discover a wide variety of previously unknown branches of solutions. We analyze the stability of the newly discovered branches and discuss the bifurcations that relate them to known solutions both in the near linear (Cartesian, as well as polar) and in the highly nonlinear regimes. While deflated continuation is not guaranteed to compute the full bifurcation diagram, this analysis is a potent demonstration that the algorithm can discover new nonlinear states and provide insights into the energy landscape of complex high-dimensional Hamiltonian dynamical systems.

nlin.PS↗

arXiv2017

Existence, Stability & Dynamics of Nonlinear Modes in a 2d Partially $\mathcal{PT}$ Symmetric Potential

It is known that multidimensional complex potentials obeying $\mathcal{PT}$-symmetry may possess all real spectra and continuous families of solitons. Recently it was shown that for multi-dimensional systems these features can persist when the parity symmetry condition is relaxed so that the potential is invariant under reflection in only a single spatial direction. We examine the existence, stability and dynamical properties of localized modes within the cubic nonlinear Schrödinger equation in such a scenario of partially $\mathcal{PT}$-symmetric potential.

nlin.PS↗

arXiv2016

On Two-Component Dark-Bright Solitons in Three-dimensional Atomic Bose-Einstein Condensates

In the present work, we revisit two-component Bose-Einstein condensates in their fully three-dimensional (3d) form. Motivated by earlier studies of dark-bright solitons in the 1d case, we explore the stability of these structures in their fully 3d form in two variants. In one the dark soliton is planar and trapping a planar bright (disk) soliton. In the other case, a dark spherical shell soliton creates an effective potential in which a bright spherical shell of atoms is trapped in the second component. We identify these solutions as numerically exact states (up to a prescribed accuracy) and perform a Bogolyubov-de Gennes linearization analysis that illustrates that both structures can be dynamically stable in suitable intervals of sufficiently low chemical potentials. We corroborate this finding theoretically by analyzing the stability via degenerate perturbation theory near the linear limit of the system. When the solitary waves are found to be unstable, we explore their dynamical evolution via direct numerical simulations which, in turn, reveal novel waveforms that are more robust. Finally, using the SO$(2)$ symmetry of the model, we produce multi-dark-bright planar or shell solitons involved in pairwise oscillatory motion.

cond-mat.quant-gas↗

arXiv2016

Dark-bright soliton interactions beyond the integrable limit

In this work we present a systematic theoretical analysis regarding dark-bright solitons and their interactions, motivated by recent advances in atomic two-component repulsively interacting Bose-Einstein condensates. In particular, we study analytically via a two-soliton ansatz adopted within a variational formulation the interaction between two dark-bright solitons in a homogeneous environment beyond the integrable regime, by considering general inter/intra-atomic interaction coefficients. We retrieve the possibility of a fixed point in the case where the bright solitons are out of phase. As the inter-component interaction is increased, we also identify an exponential instability of the two-soliton state, associated with a subcritical pitchfork bifurcation. The latter gives rise to an asymmetric partition of the bright soliton mass and dynamically leads to spontaneous splitting of the bound pair. In the case of the in-phase bright solitons, we explain via parsing the analytical approximations and monitoring the direct dynamics why no such pair is identified, despite its prediction by the variational analysis.

cond-mat.quant-gas↗

arXiv2016

Nonlinear Coherent Structures in Granular Crystals

The study of granular crystals, metamaterials that consist of closely packed arrays of particles that interact elastically, is a vibrant area of research that combines ideas from disciplines such as materials science, nonlinear dynamics, and condensed-matter physics. Granular crystals, a type of nonlinear metamaterial, exploit geometrical nonlinearities in their constitutive microstructure to produce properties (such as tunability and energy localization) that are not conventional to engineering materials and linear devices. In this topical review, we focus on recent experimental, computational, and theoretical results on nonlinear coherent structures in granular crystals. Such structures --- which include traveling solitary waves, dispersive shock waves, and discrete breathers --- have fascinating dynamics, including a diversity of both transient features and robust, long-lived patterns that emerge from broad classes of initial data. In our review, we primarily discuss phenomena in one-dimensional crystals, as most research has focused on such scenarios, but we also present some extensions to two-dimensional settings. Throughout the review, we highlight open problems and discuss a variety of potential engineering applications that arise from the rich dynamic response of granular crystals.

nlin.PS↗

arXiv2016

From Solitons to Rogue Waves in Nonlinear Left-Handed Metamaterials

In the present work, we explore soliton and rogue-like wave solutions in the transmission line analogue of a nonlinear left-handed metamaterial. The nonlinearity is expressed through a voltagedependent and symmetric capacitance motivated by the recently developed ferroelectric barium strontium titanate (BST) thin film capacitor designs. We develop both the corresponding nonlinear dynamical lattice, as well as its reduction via a multiple scales expansion to a nonlinear Schrödinger (NLS) model for the envelope of a given carrier wave. The reduced model can feature either a focusing or a defocusing nonlinearity depending on the frequency (wavenumber) of the carrier. We then consider the robustness of different types of solitary waves of the reduced model within the original nonlinear left-handed medium. We find that both bright and dark solitons persist in a suitable parametric regime, where the reduction to the NLS is valid. Additionally, for suitable initial conditions, we observe a rogue wave type of behavior, that differs significantly from the classic Peregrine rogue wave evolution, including most notably the breakup of a single Peregrine-like pattern into solutions with multiple wave peaks. Finally, we touch upon the behavior of generalized members of the family of the Peregrine solitons, namely Akhmediev breathers and Kuznetsov-Ma solitons, and explore how these evolve in the left-handed transmission line.

nlin.PS↗

arXiv2016

To Infinity and Beyond: Some ODE and PDE Case Studies

When mathematical/computational problems reach infinity, extending analysis and/or numerical computation beyond it becomes a notorious challenge. We suggest that, upon suitable singular transformations (that can in principle be computationally detected on the fly) it becomes possible to go beyond infinity to the other side, with the solution becoming again well behaved and the computations continuing normally. In our lumped, Ordinary Differential Equation (ODE) examples this infinity crossing can happen instantaneously; at the spatially distributed, Partial Differential Equation (PDE) level the crossing of infinity may even persist for finite time, necessitating the introduction of conceptual (and computational) buffer zones in which an appropriate singular transformation is continuously (locally) detected and performed. These observations (and associated tools) could set the stage for a systematic approach to bypassing infinity (and thus going beyond it) in a broader range of evolution equations; they also hold the promise of meaningfully and seamlessly performing the relevant computations. Along the path of our analysis, we present a regularization process via complexification and explore its impact on the dynamics; we also discuss a set of compactification transformations and their intuitive implications.

math.DS↗

arXiv2016

Rogue Waves in Ultracold Bosonic Seas

In this work, we numerically consider the initial value problem for nonlinear Schrödinger (NLS) type models arising in the physics of ultracold boson gases, with generic Gaussian wavepacket initial data. The corresponding Gaussian's width and, wherever relevant also its amplitude, serve as control parameters. First we explore the one-dimensional, standard NLS equation with general power law nonlinearity, in which large amplitude excitations reminiscent of Peregrine solitons or regular solitons appear to form, as the width of the relevant Gaussian is varied. Furthermore, the variation of the nonlinearity exponent aims at a first glimpse of the interplay between rogue or soliton formation and collapse features. The robustness of the main features to noise in the initial data is also confirmed. To better connect our study with the physics of atomic condensates, and explore the role of dimensionality effects, we also consider the nonpolynomial Schrödinger equation (NPSE), as well as the full three-dimensional NLS equation, and examine the degree to which relevant considerations generalize.

nlin.PS↗

arXiv2016

Vortex-soliton complexes in coupled nonlinear Schrödinger equations with unequal dispersion coefficients

We consider a two-component, two-dimensional nonlinear Schrödinger system with unequal dispersion coefficients and self-defocusing nonlinearities, chiefly with equal strengths of the self- and cross-interactions. In this setting, a natural waveform with a nonvanishing background in one component is a vortex, which induces an effective potential well in the second component, via the nonlinear coupling of the two components. We show that the potential well may support not only the fundamental bound state, but also multi-ring excited radial state complexes for suitable ranges of values of the dispersion coefficient of the second component. We systematically explore the existence, stability, and nonlinear dynamics of these states. The complexes involving the excited radial states are weakly unstable, with a growth rate depending on the dispersion of the second component. Their evolution leads to transformation of the multi-ring complexes into stable VB solitons ones with the fundamental state in the second component. The excited states may be stabilized by a harmonic-oscillator trapping potential, as well as by unequal strengths of the self- and cross-repulsive nonlinearities.

nlin.PS↗

arXiv2016

The emergence and analysis of Kuramoto-Sakaguchi-like models as an effective description for the dynamics of coupled Wien-bridge oscillators

We derive the Kuramoto-Sakaguchi model from the basic circuit equations governing two coupled Wien-bridge oscillators. A Wien-bridge oscillator is a particular realization of a tunable autonomous oscillator that makes use of frequency filtering (via a RC band-pass filter) and positive feedback (via an Op-Amp). In the last few years, such oscillators have started to be utilized in synchronization studies. We first show that the Wien-bridge circuit equations can be cast in the form of a coupled pair of Duffing - Van der Pol equations. Subsequently, by applying the method of multiple time scales, we derive the differential equations that govern the slow evolution of the oscillator phases and amplitudes. These equations are directly reminiscent of the Kuramoto-Sakaguchi type models for the study of synchronization. We analyze the resulting system in terms of existence and stability of various coupled oscillator solutions and explain on that basis how their synchronization emerges. The phase-amplitude equations are also compared numerically to the original circuit equations, and good agreement is found. Finally, we report on experimental measurements on two coupled Wien-bridge oscillators and relate the results back to the theoretical predictions.

nlin.PS↗

arXiv2016

The dynamical playground of a higher-order cubic Ginzburg-Landau equation: from orbital connections and limit cycles to invariant tori and the onset of chaos

The dynamical behavior of a higher-order cubic Ginzburg-Landau equation is found to include a wide range of scenarios due to the interplay of higher-order physically relevant terms. We find that the competition between the third-order dispersion and stimulated Raman scattering effects, gives rise to rich dynamics: this extends from Poincaré-Bendixson--type scenarios, in the sense that bounded solutions may converge either to distinct equilibria via orbital connections, or space-time periodic solutions, to the emergence of almost periodic and chaotic behavior. One of our main results is that the third-order dispersion has a dominant role in the development of such complex dynamics, since it can be chiefly responsible (i.e., even in the absence of the other higher-order effects) for the existence of the periodic, quasi-periodic and chaotic spatiotemporal structures. Suitable low-dimensional phase space diagnostics are devised and used to illustrate the different possibilities and identify their respective parametric intervals over multiple parameters of the model.

nlin.PS↗

arXiv2016

Vector Dark-Antidark Solitary Waves in Multi-Component Bose-Einstein condensates

Multi-component Bose-Einstein condensates exhibit an intriguing variety of nonlinear structures. In recent theoretical work, the notion of magnetic solitons has been introduced. Here we generalize this concept to vector dark-antidark solitary waves in multi-component Bose-Einstein condensates. We first provide concrete experimental evidence for such states in an atomic BEC and subsequently illustrate the broader concept of these states, which are based on the interplay between miscibility and inter-component repulsion. Armed with this more general conceptual framework, we expand the notion of such states to higher dimensions presenting the possibility of both vortex-antidark states and ring-antidark-ring (dark soliton) states. We perform numerical continuation studies, investigate the existence of these states and examine their stability using the method of Bogolyubov-de Gennes analysis. Dark-antidark and vortex-antidark states are found to be stable for broad parametric regimes. In the case of ring dark solitons, where the single-component ring state is known to be unstable, the vector entity appears to bear a progressively more and more stabilizing role as the inter-component coupling is increased.

cond-mat.quant-gas↗

arXiv2016

On the spontaneous time-reversal symmetry breaking in synchronously-pumped passive Kerr resonators

We study the spontaneous temporal symmetry breaking instability in a coherently-driven passive optical Kerr resonator observed experimentally by Xu and Coen in Opt.~Lett.~{\bf 39}, 3492 (2014). We perform a detailed stability analysis of the Lugiato-Lefever model for the optical Kerr resonators and analyze the temporal bifurcation structure of stationary symmetric and the emerging asymmetric states as a function of the pump power. For intermediate pump powers a pitchfork loop is responsible for the destabilization of symmetric states towards stationary asymmetric ones while at large pump powers we find the emergence of periodic asymmetric solutions via a Hopf bifurcation. From a theoretical perspective, we use local bifurcation theory in order to analyze the most unstable eigenmode of the system. We also explore a non-conservative variational approximation capturing, among others, the evolution of the solution's amplitude, width and center of mass. Both methods provide insight towards the pitchfork bifurcations associated with the symmetry breaking.

nlin.PS↗