Discrete Gap Solitons In A Diffraction-Managed Waveguide Array, 2011 University of Massachusetts - Amherst

#### Discrete Gap Solitons In A Diffraction-Managed Waveguide Array, Pg Kevrekidis

*Panos Kevrekidis*

A model including two nonlinear chains with linear and nonlinear couplings between them, and opposite signs of the discrete diffraction inside the chains, is introduced. In the case of the cubic [ χ(3)] nonlinearity, the model finds two different interpretations in terms of optical waveguide arrays, based on the diffraction-management concept. A continuum limit of the model is tantamount to a dual-core nonlinear optical fiber with opposite signs of dispersions in the two cores. Simultaneously, the system is equivalent to a formal discretization of the standard model of nonlinear optical fibers equipped with the Bragg grating. A straightforward discrete second-harmonic-generation ...

Localized Breathing Modes In Granular Crystals With Defects, 2011 University of Massachusetts - Amherst

#### Localized Breathing Modes In Granular Crystals With Defects, G Theocharis, M Kavousanakis, Pg Kevrekidis, C Daraio, Ma Porter, Ig Kevrekidis

*Panos Kevrekidis*

We study localized modes in uniform one-dimensional chains of tightly packed and uniaxially compressed elastic beads in the presence of one or two light-mass impurities. For chains composed of beads of the same type, the intrinsic nonlinearity, which is caused by the Hertzian interaction of the beads, appears not to support localized, breathing modes. Consequently, the inclusion of light-mass impurities is crucial for their appearance. By analyzing the problem’s linear limit, we identify the system’s eigenfrequencies and the linear defect modes. Using continuation techniques, we find the solutions that bifurcate from their linear counterparts and study their linear ...

Static And Rotating Domain-Wall Cross Patterns In Bose-Einstein Condensates, 2011 University of Massachusetts - Amherst

#### Static And Rotating Domain-Wall Cross Patterns In Bose-Einstein Condensates, B Malomed, H Nistazakis, D Frantzekakis, Pg Kevrekidis

*Panos Kevrekidis*

For a Bose-Einstein condensate (BEC) in a two-dimensional (2D) trap, we introduce cross patterns, which are generated by the intersection of two domain walls (DWs) separating immiscible species, with opposite signs of the wave functions in each pair of sectors filled by the same species. The cross pattern remains stable up to the zero value of the immiscibility parameter ∣Δ∣, while simpler rectilinear (quasi-1D) DWs exist only for values of ∣Δ∣ essentially exceeding those in BEC mixtures (two spin states of the same isotope) currently available to the experiment. Both symmetric and asymmetric cross configurations are investigated, with equal or ...

Asymptotic Stability Of Small Solitons In The Discrete Nonlinear Schr¨Odinger Equation In One Dimension, 2011 University of Massachusetts - Amherst

#### Asymptotic Stability Of Small Solitons In The Discrete Nonlinear Schr¨Odinger Equation In One Dimension, Pg Kevrekidis

*Panos Kevrekidis*

Asymptotic stability of small solitons in one dimension is proved in the framework of a discrete nonlinear Schr¨odinger equation with septic and higher power-law nonlinearities and an external potential supporting a simple isolated eigenvalue. The analysis relies on the dispersive decay estimates from Pelinovsky & Stefanov (2008) and the arguments of Mizumachi (2008) for a continuous nonlinear Schr¨odinger equation in one dimension. Numerical simulations suggest that the actual decay rate of perturbations near the asymptotically stable solitons is higher than the one used in the analysis.

Mobility Of Discrete Solitons In Quadratically Nonlinear Media, 2011 University of Massachusetts - Amherst

#### Mobility Of Discrete Solitons In Quadratically Nonlinear Media, H Susanto, Pg Kevrekidis, R Carretero-Gonzalez, Ba Malomed, Dj Frantzeskakis

*Panos Kevrekidis*

We study the mobility of solitons in lattices with quadratic (χ(2), alias second-harmonic-generating) nonlinearity. Using the notion of the Peierls-Nabarro potential and systematic numerical simulations, we demonstrate that, in contrast with their cubic (χ(3)) counterparts, the discrete quadratic solitons are mobile not only in the one-dimensional (1D) setting, but also in two dimensions (2D), in any direction. We identify parametric regions where an initial kick applied to a soliton leads to three possible outcomes: staying put, persistent motion, or destruction. On the 2D lattice, the solitons survive the largest kick and attain the largest speed along the diagonal ...

Using The Continuous Spectrum To “Feel” Integrability: The Effect Of Boundary Conditions, 2011 University of Massachusetts - Amherst

#### Using The Continuous Spectrum To “Feel” Integrability: The Effect Of Boundary Conditions, Pg Kevrekidis

*Panos Kevrekidis*

The scope of this work is to propose a method for testing the integrability of a model partial differential (PDE) and/or differential difference equation (DDE). For monoparametric families of PDE/DDE’s, that are known to possess isolated integrable points, we find that very special features occur in the continuous (“phonon”) spectrum at these “singular” points. We identify these features in the case example of a PDE and a DDE (that sustain front and pulse-like solutions respectively) for different types of boundary conditions. The key finding of the work is that such spectral features are generic near the singular ...

Lyapunov-Schmidt Reduction Algorithm For Three-Dimensional Discrete Vortices, 2011 University of Massachusetts - Amherst

#### Lyapunov-Schmidt Reduction Algorithm For Three-Dimensional Discrete Vortices, M Lukas, D Pelinovsky, Pg Kevrekidis

*Panos Kevrekidis*

We address the persistence and stability of three-dimensional vortex configurations in the discrete nonlinear Schrödinger 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 lattice and applied to the onsite and offsite configurations (called the vortex cross and the vortex cell) by ...

From Feshbach-Resonance Managed Bose-Einstein Condensates To Anisotropic Universes: Applications Of The Ermakov-Pinney Equation With Time-Dependent Nonlinearity, 2011 University of Massachusetts - Amherst

#### From Feshbach-Resonance Managed Bose-Einstein Condensates To Anisotropic Universes: Applications Of The Ermakov-Pinney Equation With Time-Dependent Nonlinearity, G Herring, Pg Kevrekidis, F Williams, T Christodoulakis, Dj Frantzeskakis

*Panos Kevrekidis*

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 ...

Standard Nearest-Neighbour Discretizations Of Klein–Gordon Models Cannot Preserve Both Energy And Linear Momentum, 2011 University of Massachusetts - Amherst

#### Standard Nearest-Neighbour Discretizations Of Klein–Gordon Models Cannot Preserve Both Energy And Linear Momentum, S V. Dmitriev, Pg Kevrekidis

*Panos Kevrekidis*

We consider nonlinear Klein–Gordon wave equations and illustrate that standard discretizations thereof (involving nearest neighbours) may preserve either standardly defined linear momentum or standardly defined total energy but not both. This has a variety of intriguing implications for the 'non-potential' discretizations that preserve only the linear momentum, such as the self-accelerating or self-decelerating motion of coherent structures such as discrete kinks in these nonlinear lattices.

Exact Static Solutions For Discrete Φ4 Models Free Of The Peierls-Nabarro Barrier: Discretized First-Integral Approach, 2011 University of Massachusetts - Amherst

#### Exact Static Solutions For Discrete Φ4 Models Free Of The Peierls-Nabarro Barrier: Discretized First-Integral Approach, S V. Dmitriev, Pg Kevrekidis

*Panos Kevrekidis*

We propose a generalization of the discrete Klein-Gordon models free of the Peierls-Nabarro barrier derived in Spreight [Nonlinearity 12, 1373 (1999)] and Barashenkov et al. [Phys. Rev. E 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 by Dmitriev et al. [J. Phys. A 38, 7617 (2005)]. We then discuss some discrete ϕ4 models free of the Peierls-Nabarro barrier and identify for them ...

Breathers On A Background: Periodic And Quasiperiodic Solutions Of Extended Discrete Nonlinear Wave Systems, 2011 University of Massachusetts - Amherst

#### Breathers On A Background: Periodic And Quasiperiodic Solutions Of Extended Discrete Nonlinear Wave Systems, Pg Kevrekidis

*Panos Kevrekidis*

In this paper we investigate the emergence of time-periodic and time quasiperiodic (sometimes infinitely long-lived and sometimes very long-lived or metastable) solutions of discrete nonlinear wave equations: discrete sine Gordon, discrete φ4 and discrete nonlinear Schrödinger equation (DNLS). The solutions we consider are periodic oscillations on a kink or standing wave breather background. The origin of these oscillations is the presence of internal modes, associated with the static ground state. Some of these modes are associated with the breaking of translational invariance, in going from a spatially continuous to a spatially discrete system. Others are associated with discrete modes which ...

Two-Soliton Collisions In A Near-Integrable Lattice System, 2011 University of Massachusetts - Amherst

#### Two-Soliton Collisions In A Near-Integrable Lattice System, S Dmitriev, Pg Kevrekidis

*Panos Kevrekidis*

We examine collisions between identical solitons in a weakly perturbed Ablowitz-Ladik (AL) model, augmented by either onsite cubic nonlinearity (which corresponds to the Salerno model, and may be realized as an array of strongly overlapping nonlinear optical waveguides) or a quintic perturbation, or both. Complex dependences of the outcomes of the collisions on the initial phase difference between the solitons and location of the collision point are observed. Large changes of amplitudes and velocities of the colliding solitons are generated by weak perturbations, showing that the elasticity of soliton collisions in the AL model is fragile (for instance, the Salerno ...

Solitons In One-Dimensional Nonlinear Schrodinger Lattices With A Local Inhomogeneity, 2011 University of Massachusetts - Amherst

#### Solitons In One-Dimensional Nonlinear Schrodinger Lattices With A Local Inhomogeneity, F Palmero, R Carretero-Gonzalez, J Cuevas, Pg Kevrekidis, W Krolikowski

*Panos Kevrekidis*

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 ...

Surface Solitons In Three Dimensions, 2011 University of Massachusetts - Amherst

#### Surface Solitons In Three Dimensions, Qe Hoq, R Carretero-Gonzalez, Pg Kevrekidis, Ba Malomed, Dj Frantzeskakis, Yv Bludov, Vv Konotop

*Panos Kevrekidis*

We study localized modes on the surface of a three-dimensional dynamical lattice. The stability of these structures on the surface is investigated and compared to that in the bulk of the lattice. Typically, the surface makes the stability region larger, an extreme example of that being the three-site “horseshoe”-shaped structure, which is always unstable in the bulk, while at the surface it is stable near the anticontinuum limit. We also examine effects of the surface on lattice vortices. For the vortex placed parallel to the surface, the increased stability-region feature is also observed, while the vortex cannot exist in ...

Discrete Skyrmions In 2+1 And 3+1 Dimensions, 2011 University of Massachusetts - Amherst

#### Discrete Skyrmions In 2+1 And 3+1 Dimensions, T Ioannidou, Pg Kevrekidis

*Panos Kevrekidis*

This Letter describes a lattice version of the Skyrme model in 2+1 and 3+1 dimensions. The discrete model is derived from a consistent discretization of the radial continuum problem. Subsequently, the existence and stability of the skyrmion solutions existing on the lattice are investigated. One consequence of the proposed models is that the corresponding discrete skyrmions have a high degree of stability, similar to their continuum counterparts.

On Some Classes Of Mkdv Periodic Solutions, 2011 University of Massachusetts - Amherst

#### On Some Classes Of Mkdv Periodic Solutions, Pg Kevrekidis

*Panos Kevrekidis*

We obtain exact periodic solutions of the positive and negative modified Kortweg–de Vries (mKdV) equations. We examine the dynamical stability of these solitary wave lattices through direct numerical simulations. While the positive mKdV breather lattice solutions are found to be unstable, the two-soliton lattice solution of the same equation is found to be stable. Similarly, a negative mKdV lattice solution is found to be stable. We also touch upon the implications of these results for the KdV equation.

Vortex Structures Formed By The Interference Of Sliced Condensates, 2011 University of Massachusetts - Amherst

#### Vortex Structures Formed By The Interference Of Sliced Condensates, R Carretero-Gonzalez, N Whitaker, Pg Kevrekidis, Dj Frantzeskakis

*Panos Kevrekidis*

We study the formation of vortices, vortex necklaces, and vortex ring structures as a result of the interference of higher-dimensional Bose-Einstein condensates (BECs). This study is motivated by earlier theoretical results pertaining to the formation of dark solitons by interfering quasi-one-dimensional BECs, as well as recent experiments demonstrating the formation of vortices by interfering higher-dimensional BECs. Here, we demonstrate the genericness of the relevant scenario, but also highlight a number of additional possibilities emerging in higher-dimensional settings. A relevant example is, e.g., the formation of a “cage” of vortex rings surrounding the three-dimensional bulk of the condensed atoms. The ...

Dynamics Of Dark–Bright Solitons In Cigar-Shaped Bose–Einstein Condensates, 2011 University of Massachusetts - Amherst

#### Dynamics Of Dark–Bright Solitons In Cigar-Shaped Bose–Einstein Condensates, S Middelkamp, J J. Chang, C Hammer, R Carretero-Gonzalez, Pg Kevrekidis

*Panos Kevrekidis*

We explore the stability and dynamics of dark–bright (DB) solitons in two-component elongated Bose–Einstein condensates by developing effective one-dimensional vector equations and solving the three-dimensional Gross–Pitaevskii equations. A strong dependence of the oscillation frequency and of the stability of the DB soliton on the atom number of its components is found; importantly, the wave may become dynamically unstable even in the 1D regime. As the atom number in the dark-soliton-supporting component is further increased, spontaneous symmetry breaking leads to oscillatory dynamics in the transverse degrees of freedom. Moreover, the interactions of two DB solitons are investigated with ...

Averaging For Solitons With Nonlinearity Management, 2011 University of Massachusetts - Amherst

#### Averaging For Solitons With Nonlinearity Management, D E. Pelinovsky, Pg Kevrekidis

*Panos Kevrekidis*

We develop an averaging method for solitons of the nonlinear Schrödinger equation with a periodically varying nonlinearity coefficient, which is used to effectively describe solitons in Bose-Einstein condensates, in the context of the recently proposed technique of Feshbach resonance management. Using the derived local averaged equation, we study matter-wave bright and dark solitons and demonstrate a very good agreement between solutions of the averaged and full equations.

Avoiding Infrared Catastrophes In Trapped Bose-Einstein Condensates, 2011 University of Massachusetts - Amherst

#### Avoiding Infrared Catastrophes In Trapped Bose-Einstein Condensates, Pg Kevrekidis, G Theocharis, Dj Frantzeskakis, A Trombettoni

*Panos Kevrekidis*

This paper is concerned with the long-wavelength instabilities (infrared catastrophes) occurring in Bose-Einstein condensates (BECs). We examine the modulational instability in “cigar-shaped” (one-dimensional) attractive BECs and the transverse instability of dark solitons in “pancake” (two-dimensional) repulsive BECs. We suggest mechanisms, and give explicit estimates, on how to engineer the trapping conditions of the condensate to avoid such instabilities: the main result being that a tight enough trapping potential suppresses the instabilities present in the homogeneous limit. We compare the obtained estimates with numerical results and we highlight the relevant regimes of dynamical behavior.