We use experiments, numerical simulations, and theoretical analysis to investigate the propagation of highly nonlinear solitary waves in periodic arrangements of dimer (two-mass) and trimer (three-mass) cell structures in one-dimensional granular lattices. To vary the composition of the fundamental periodic units in the granular chains, we utilize beads of different materials (stainless steel, bronze, glass, nylon, polytetrafluoroethylene, and rubber). This selection allows us to tailor the response of the system based on the masses, Poisson ratios, and elastic moduli of the components. For example, we examine dimer configurations with two types of heavy particles, two types of light particles, and alternating light and heavy particles. Employing a model with Hertzian interactions between adjacent beads, we find very good agreement between experiments and numerical simulations. We find equally good agreement between these results and a theoretical analysis of the model in the long-wavelength regime that we derive for heterogeneous environments (dimer chains) and general bead interactions. Our analysis encompasses previously-studied examples as special cases and also provides key insights on the influence of heterogeneous lattices on the properties (width and propagation speed) of the nonlinear wave solutions of this system.
We focus on solitary waves generated in arrays of lightly contacting spherical elastic granules by shock forces of steep rise and slow decay durations, and establish a priori: (i) whether the peak value of the resulting solitary wave would be greater, equal, or less than the peak value of the input shock force; (ii) the magnitude of the peak value of the solitary waves; (iii) the magnitude of the linear momentum in each solitary wave; (iv) the magnitude of the linear momentum added to the remaining granules, if the first granule is ejected; and (v) a quantitative estimate of the effect of the …
We provide a quantitative characterization of dissipative effects in one-dimensional granular crystals. We use the propagation of highly nonlinear solitary waves as a diagnostic tool and develop optimization schemes that allow one to compute the relevant exponents and prefactors of the dissipative terms in the equations of motion. We thereby propose a quantitatively-accurate extension of the Hertzian model that encompasses dissipative effects via a discrete Laplacian of the velocities. Experiments and computations with steel, brass, and polytetrafluoroethylene reveal a {\em common} dissipation…
We report the propagation of highly nonlinear solitary waves in heterogeneous, periodic granular media using experiments, numerical simulations, and theoretical analysis. We examine periodic arrangements of particles in experiments in which stiffer/heavier beads (stainless steel) are alternated with softer/lighter ones (PTFE beads). We find excellent agreement between experiments and numerics in a model with Hertzian interactions between adjacent beads, which in turn agrees very well with a theoretical analysis of the model in the long-wavelength regime that we derive for heterogeneous environ…
Solitary waves (SWs) are generated in monoatomic (homogeneous) lightly contacting spherical granules by an applied input force of any time-variation and intensity. We consider finite duration shock loads and focus on the transition regime that leads to the formation of SWs. Based on geometrical and material properties of the granules and the properties of the input shock, we provide explicit analytic expressions to calculate the peak value of the compressive contact force at each contact point in the transition regime that precedes the formation of a primary solitary wave. We also provide expl…
We study ``nanoptera'', which are non-localized solitary waves with exponentially small but non-decaying oscillations, in two singularly-perturbed Hertzian chains with precompression. These two systems are woodpile chains (which we model as systems of Hertzian particles and springs) and diatomic Hertzian chains with alternating masses. We demonstrate that nanoptera arise from Stokes phenomena and appear as special curves, called Stokes curves, are crossed in the complex plane. We use techniques from exponential asymptotics to obtain approximations of the oscillation amplitudes. Our analysis de…
We demonstrate experimentally the existence of three-wave resonant interaction solitary triplets in quadratic media. Stable velocity-locked bright-dark-bright spatial solitary triplets, determined by the balance between the energy exchange rates and the velocity mismatch between the interacting waves, are excited in a KTP crystal.
We consider the wave propagation in media with step-like inhomogeneities. We choose two different protocols: (I) a step-like spatial dependence of the coupling constant that physically corresponds to the junction of two systems and (II) the step-like time-dependence of the coupling constant that corresponds to a quench protocol. In both scenarios, we study the propagation of the solitary wave in $φ^4$ and the sine-Gordon model. Due to topological protection, the solitary wave retains its form. This allows us to give model-independent predictions about its final velocity, which can be interpret…
We study solitary wave propagation in 1D granular crystals with Hertz-like interaction potentials. We consider interfaces between media with different exponents in the interaction potential. For an interface with increasing interaction potential exponent along the propagation direction we obtain mainly transmission with delayed secondary transmitted and reflected pulses. For interfaces with decreasing interaction potential exponent we observe both significant reflection and transmission of the solitary wave, where the transmitted part of the wave forms a multipulse structure. We also investiga…
A new class of solitary waves arises in the solution of nonlinear wave equations with constant impedance and no dispersive terms. They depend on a balance between nonlinearity and a dispersion-like effect due to spatial variation in the sound speed of the medium. A high-order homogenized model confirms this effective dispersive behavior and its solutions agree well with those obtained by direct simulation of the variable-coefficient system. These waves are observed to be long-time stable, globally attracting solutions that arise in general as solutions to nonlinear wave problems with periodica…
Solitary waves in one-dimensional periodic media are discussed employing the nonlinear Schrödinger equation with a spatially periodic potential as a model. This equation admits two families of gap solitons that bifurcate from the edges of Bloch bands in the linear wave spectrum. These fundamental solitons may be positioned only at specific locations relative to the potential; otherwise, they become nonlocal owing to the presence of growing tails of exponentially-small amplitude with respect to the wave peak amplitude. Here, by matching the tails of such nonlocal solitary waves, higher-order lo…
Solitary waves bifurcated from edges of Bloch bands in two-dimensional periodic media are determined both analytically and numerically in the context of a two-dimensional nonlinear Schrödinger equation with a periodic potential. Using multi-scale perturbation methods, envelope equations of solitary waves near Bloch bands are analytically derived. These envelope equations reveal that solitary waves can bifurcate from edges of Bloch bands under either focusing or defocusing nonlinearity, depending on the signs of second-order dispersion coefficients at the edge points. Interestingly, at edge poi…
We report exact bright and dark soliton solution to the nonlinear evolution equation derived by Moses and Wise [Phys. Rev. Lett. 97, 073903, (2006)] for cascaded quadratic media beyond the slowly varying envelope approximations. The integrability aspects of the model are addressed. The traveling wave hypothesis as well as the ansatz method is employed to obtain an exact 1-soliton solution. Both bright and dark soliton solutions are obtained. The corresponding constraint conditions are obtained in order for the soliton solutions to exist.
We study the propagation of an unusual type of periodic travelling waves in chains of identical beads interacting via Hertz's contact forces. Each bead periodically undergoes a compression phase followed by a free flight, due to special properties of Hertzian interactions (fully nonlinear under compression and vanishing in the absence of contact). We prove the existence of such waves close to binary oscillations, and numerically continue these solutions when their wavelength is increased. In the long wave limit, we observe their convergence towards shock profiles consisting of small compressio…
We report on the strongly nonlinear dynamics of an array of weakly coupled, non-compressed, parallel granular chains subject to a local initial impulse. The motion of the granules in each chain is constrained to be in one direction which coincides with the orientation of the chain. We show that in spite of the fact that the applied impulse is applied to a one of the granular chains, the resulting pulse that initially propagating only in the excited chain gets gradually equi-partitioned between its neighboring chains, and eventually in all chains of the array. In particular, the initially stron…
Solitons are long-lived wavepackets that propagate without dispersion and exist in a wide range of one-dimensional (1D) nonlinear systems. A Bose-Einstein condensate trapped in a quasi-1D waveguide can support bright-solitary-matter waves (3D analogues of solitons) when interatomic interactions are sufficiently attractive that they cancel dispersion. Solitary-matter waves are excellent candidates for a new generation of highly sensitive interferometers, as their non-dispersive nature allows them to acquire phase shifts for longer times than conventional matter-waves interferometers. However, s…
We consider the interplay between nonlocal nonlinearity and randomness for two different nonlinear Schrödinger models. We show that stability of bright solitons in presence of random perturbations increases dramatically with the nonlocality-induced finite correlation length of the noise in the transverse plane, by means of both numerical simulations and analytical estimates. In fact, solitons are practically insensitive to noise when the correlation length of the noise becomes comparable to the extent of the wave packet. We characterize soliton stability using two different criteria based on t…
In this paper, we examine in detail the principal branches of solutions that arise in vector discrete models with nonlinear inter-component coupling and four wave mixing. The relevant four branches of solutions consist of two single mode branches (transverse electric and transverse magnetic) and two mixed mode branches, involving both components (linearly polarized and elliptically polarized). These solutions are obtained explicitly and their stability is analyzed completely in the anti-continuum limit (where the nodes of the lattice are uncoupled), illustrating the supercritical pitchfork nat…
We present a wide class of novel solitary and periodic waves in a non-centrosymmetric waveguide exhibiting second- and third-order nonlinearities. We show the existence of bright, gray, and W-shaped solitary waves as well as periodic waves for extended nonlinear Schrödinger equation with quadratic and cubic nonlinearities. We also obtained the exact analytical algebraic-type solitary waves of the governing equation, including bright and W-shaped waves. The results illustrate the propagation of potentially rich set of nonlinear structures through the optical waveguiding media. Such privileged w…
One of the most interesting phenomena occuring in nonlinear media models is the existence of wave patterns, such as kinks, solitons, compactons, peakons and many others. There are known numerous nonlinear evolutionary PDEs, supporting soliton (multi-soliton) and compacton traveling wave (TW) solutions. Unfortunately, the vast majority of the models, with the exception of completely integrable ones, do not enable to analyze the properties of solitary waves interaction using only qualitative methods. Therefore it is instructive, when dealing with the non-integrable PDEs, to combine the qualitati…
We demonstrate the existence of two species of stable bright solitons, fundamental and dipole ones, in one-dimensional self-defocusing nonlocal media, with the local value of nonlinearity coefficient having one or several minima and growing at any rate faster than |x| at large values of coordinate x. The model can be derived for a slab optical waveguide with thermal nonlinearity. The most essential difference from the local counterpart of this system is the competition between two different spatial scales, the one determining the modulation pattern of the nonlinearity coefficient, and the corr…
We demonstrate the possibility of self-trapping of optical beams by use of quasi phase matching in a counterpropagating configuration in quadratic media. We also show the predominant stability of these spatial self-guided beams and estimate the power level required for their experimental observation.
In the present work, we develop a systematic examination of the existence, stability and dynamical properties of a discrete breather at the interface between a diatomic and a monoatomic granular chain. We remarkably find that such an "interface breather" is more robust than its bulk diatomic counterpart throughout the gap of the linear spectrum. The latter linear spectral gap needs to exist for the breather state to arise and the relevant spectral conditions are discussed. We illustrate the minimal excitation conditions under which such an interface breather can be "nucleated" and analyze its …
We introduce the simplest one-dimensional model of a dispersive optical medium with saturable dissipative nonlinearity and filtering (dispersive loss) which gives rise to stable solitary pulses (autosolitons). In the particular case when the dispersive loss is absent, the same model may also be interpreted as describing a stationary field in a planar optical waveguide with uniformly distributed saturable gain and absorption. In a certain region of the model's parameter space, two coexisting solitary-pulse solutions are found numerically, one of which may be stable. Solving the corresponding li…
We study a system of nonlinear differential equations simulating transport phenomena in active media. The model we are interested in is a generalization of the celebrated FitzHugh-Nagumo system, describing the nerve impulse propagation in axon. The modeling system is shown to possesses soliton-like solutions under certain restrictions on the parameters. The results of theoretical studies are backed by the direct numerical simulation.
We consider the question of existence of "bell-shaped" (i.e. non-increasing for x>0 and non-decreasing for x<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 a…
We present a detailed analysis of the modulational instability (MI) of ground-state bright solitary solutions of two incoherently coupled nonlinear Schrödinger equations. Varying the relative strength of cross-phase and self-phase effects we show existence and origin of four branches of MI of the two-wave solitary solutions. We give a physical interpretation of our results in terms of the group velocity dispersion (GVD) induced polarization dynamics of spatial solitary waves. In particular, we show that in media with normal GVD spatial symmetry breaking changes to polarization symmetry breakin…
We demonstrate that weak parametric interaction of a fundamental beam with its third harmonic field in Kerr media gives rise to a rich variety of families of non-fundamental (multi-humped) solitary waves. Making a comprehensive comparison between bifurcation phenomena for these families in bulk media and planar waveguides, we discover two novel types of soliton bifurcations and other interesting findings. The later includes (i) multi-humped solitary waves without even or odd symmetry and (ii) multi-humped solitary waves with large separation between their humps which, however, may not be viewe…
We obtain a class of elliptic wave solutions of coupled nonlinear Helmholtz (CNLH) equations describing nonparaxial ultra-broad beam propagation in nonlinear Kerr-like media, in terms of the Jacobi elliptic functions and also discuss their limiting forms (hyperbolic solutions). Especially, we show the existence of non-trivial solitary wave profiles in the CNLH system. The effect of nonparaxiality on the speed, pulse width and amplitude of the nonlinear waves is analysed in detail. Particularly a mechanism for tuning the speed by altering the nonparaxial parameter is proposed. We also identify …
We derive a model for the propagation of short pulses in nonlinear media. The model is a higher order regularization of the short pulse equation (SPE). The regularization term arises as the next term in the expansion of the susceptibility in derivation of the SPE. Without the regularization term there do not exist traveling pulses in the class of piecewise smooth functions with one discontinuity. However, when the regularization term is added we show, for a particular parameter regime, that the equation supports smooth traveling waves which have structure similar to solitary waves of the modif…
Our aim in the present work is to develop approximations for the collisional dynamics of traveling waves in the context of granular chains in the presence of precompression. To that effect, we aim to quantify approximations of the relevant Hertzian FPU-type lattice through both the Korteweg-de Vries (KdV) equation and the Toda lattice. Using the availability in such settings of both 1-soliton and 2-soliton solutions in explicit analytical form, we initialize such coherent structures in the granular chain and observe the proximity of the resulting evolution to the underlying integrable (KdV or …
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, computationa…
This paper considers the propagation of shallow-water solitary and nonlinear periodic waves over a gradual slope with bottom friction in the framework of a variable-coefficient Korteweg-de Vries equation. We use the Whitham averaging method, using a recent development of this theory for perturbed integrable equations. This general approach enables us not only to improve known results on the adiabatic evolution of isolated solitary waves and periodic wave trains in the presence of variable topography and bottom friction, modeled by the Chezy law, but also importantly, to study the effects of th…
