Three-Dimensional Solitary Waves and Vortices in a Discrete Nonlinear Schrödinger Lattice

In a benchmark dynamical-lattice model in three dimensions, the discrete nonlinear Schrödinger 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 and may 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

Three-Dimensional Solitary Waves and Vortices in a Discrete Nonlinear Schrödinger Lattice

In a benchmark dynamical-lattice model in three dimensions, the discrete nonlinear Schrödinger 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 and may 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

Nonlinear Waves in Lattices: Past, Present, Future

In the present work, we attempt a brief summary of various areas where non-linear waves have been emerging in the phenomenology of lattice dynamical systems. These areas include non-linear optics, atomic physics, mechanical systems, electrical lattices, non-linear metamaterials, plasma dynamics and granular crystals. We give some of the recent developments in each one of these areas and speculate on some of the potentially interesting directions for future study

Highly Nonlinear Wave Propagation in Elastic Woodpile Periodic Structures

In the present work, we experimentally implement, numerically compute with, and theoretically analyze a configuration in the form of a single column woodpile periodic structure. Our main finding is that a Hertzian, locally resonant, woodpile lattice offers a test bed for the formation of genuinely traveling waves composed of a strongly localized solitary wave on top of a small amplitude oscillatory tail. This type of wave, called a nanopteron, is not only motivated theoretically and numerically, but is also visualized experimentally by means of a laser Doppler vibrometer. This system can also be useful for manipulating stress waves at will, for example, to achieve strong attenuation and modulation of high-amplitude impacts without relying on damping in the system

Multi-Component Nonlinear Waves in Optics and Atomic Condensates: Theory, Computations and Experiments

Motivated by work in nonlinear optics, as well as more recently in Bose-Einstein condensate mix- tures, we will explore a series of nonlinear states that arise in such systems. We will start from a single structure, the so-called dark-bright solitary wave, and then expand our considerations to multiple such waves, their spectral properties, nonlinear interactions and experimental observa- tions. A twist will be to consider the dark solitons of the one component as effective potentials that will trap the bright waves of the second component, an approach that will also prove useful in characterizing the bifurcations and instabilities of the system. Beating so-called dark-dark soliton variants of such states will also be touched upon. Generalizations of all these notions in higher dimensions and, so-called, vortex-bright solitons will also be offered and challenges for future work will be discussed.Non UBCUnreviewedAuthor affiliation: University of Massachusetts AmherstFacult

On 3+1 Dimensional Scalar Field Cosmologies

In this communication, we analyze the case of 3+1 dimensional scalar field cosmologies in the presence, as well as in the absence of spatial curvature, in isotropic, as well as in anisotropic settings. Our results extend those of Hawkins and Lidsey [Phys. Rev. D {\bf 66}, 023523 (2002)], by including the non-flat case. The Ermakov-Pinney methodology is developed in a general form, allowing through the converse results presented herein to use it as a tool for constructing new solutions to the original equations. As an example of this type a special blowup solution recently obtained in Christodoulakis {\it et al.} [gr-qc/0302120] is retrieved. Additional solutions of the 3+1 dimensional gravity coupled with the scalar field are also obtained. To illustrate the generality of the approach, we extend it to the anisotropic case of Bianchi types I and V and present some related open problems

Solitons and Vortices in Two-Dimensional Discrete Nonlinear Schrodinger Systems with Spatially Modulated Nonlinearity

We consider a two-dimensional (2D) generalization of a recently proposed model [Gligorić et al., Phys. Rev. E 88, 032905 (2013)], which gives rise to bright discrete solitons supported by the defocusing nonlinearity whose local strength grows from the center to the periphery. We explore the 2D model starting from the anticontinuum (AC) limit of vanishing coupling. In this limit, we can construct a wide variety of solutions including not only single-site excitations, but also dipole and quadrupole ones. Additionally, two separate families of solutions are explored: the usual “extended” unstaggered bright solitons, in which all sites are excited in the AC limit, with the same sign across the lattice (they represent the most robust states supported by the lattice, their 1D counterparts being those considered as 1D bright solitons in the above-mentioned work), and the vortex cross, which is specific to the 2D setting. For all the existing states, we explore their stability (also analytically, when possible). Typical scenarios of instability development are exhibited through direct simulations

PT-symmetric oligomers: Analytical solutions, linear stability, and nonlinear dynamics

In the present work we focus on the case of (few-site) configurations respecting the parity-time (PT) symmetry, i.e., with a spatially odd gain-loss profile. We examine the case of such “oligomers” with not only two sites, as in earlier works, but also the cases of three and four sites. While in the former case of recent experimental interest the picture of existing stationary solutions and their stability is fairly straightforward, the latter cases reveal a considerable additional complexity of solutions, including ones that exist past the linear PT-symmetry breaking point in the case of the trimer, and symmetry-breaking bifurcations, as well as more complex, even asymmetric solutions in the case of the quadrimer with nontrivial properties in their linear stability and in their nonlinear dynamics. The linearization around the obtained solutions and their dynamical evolution, when unstable, are discussed

On the existence of solitary traveling waves for generalized Hertzian chains

We consider the question of existence of “bell-shaped” (i.e. non-increasing for x \u3e 0 and non-decreasing for x \u3c 0) traveling waves for the strain variable of the generalized Hertzian model describing, in the special case of a p = 3/2 exponent, the dynamics of a granular chain. The proof of existence of such waves is based on the English and Pego [Proceedings of the AMS 133, 1763 (2005)] formulation of the problem. More specifically, we construct an appropriate energy functional, for which we show that the constrained minimization problem over bell-shaped entries has a solution. We also provide an alternative proof of the Friesecke-Wattis result [Comm. Math. Phys 161, 394 (1994)], by using the same approach (but where the minimization is not constrained over bell-shaped curves). We briefly discuss and illustrate numerically the implications on the doubly exponential decay properties of the waves, as well as touch upon the modifications of these properties in the presence of a finite precompression force in the model