Nonlinear Localized Modes Research Articles

Nonlinear localized eigenmodes for a Bose–Einstein condensate in a double-well potential

In this paper, we investigate the nonlinear localized eigenmodes for a Bose–Einstein condensate in a double-well potential. For a specific choice of the potential parameters, certain exact analytical solutions for nonlinear localized eigenmodes are presented. By applying the linear stability analysis, the stability regions of these exact nonlinear localized eigenmodes are obtained numerically. It is shown that under certain conditions, the unstable nonlinear localized modes display the breathing behavior characterized by repeated appearance of symmetric and asymmetric distributions in the two potentials. This breathing behavior is shown to arise from the symmetry breaking for these nonlinear localized eigenmodes.

Compactification tuning for nonlinear localized modes in sawtooth lattices.

We discuss the properties of nonlinear localized modes in sawtooth lattices, in the framework of a discrete nonlinear Schrödinger model with general on-site nonlinearity. Analytic conditions for existence of exact compact three-site solutions are obtained, and explicitly illustrated for the cases of power-law (cubic) and saturable nonlinearities. These nonlinear compact modes appear as continuations of linear compact modes belonging to a flat dispersion band. While for the linear system a compact mode exists only for one specific ratio of the two different coupling constants, nonlinearity may lead to compactification of otherwise noncompact localized modes for a range of coupling ratios, at some specific power. For saturable lattices, the compactification power can be tuned by also varying the nonlinear parameter. Introducing different on-site energies and anisotropic couplings yields further possibilities for compactness tuning. The properties of strongly localized modes are investigated numerically for cubic and saturable nonlinearities, and in particular their stability over large parameter regimes is shown. Since the linear flat band is isolated, its compact modes may be continued into compact nonlinear modes both for focusing and defocusing nonlinearities. Results are discussed in relation to recent realizations of sawtooth photonic lattices.

Discrete solitons in graphene metamaterials

We study nonlinear properties of multilayer metamaterials created by graphene sheets separated by dielectric layers. We demonstrate that such structures can support localized nonlinear modes described by the discrete nonlinear Schr\"{o}dinger equation and that its solutions are associated with stable discrete plasmon solitons. We also analyze the nonlinear surface modes in truncated graphene metamaterials being a nonlinear analog of surface Tamm states.

Effect of the interatomic potential stiffness on the properties of gap discrete breathers in 2D biatomic Morse crystal

Discrete breathers (DB) are spatially localized nonlinear vibrational modes in discrete lattices. The latest trend is the study of DB in various crystals. Many studies have been done recently for two-dimensional (2D) model of a crystal with Morse interatomic potential. Two types of DB have been discussed for Morse crystals, the gap DB with soft type of nonlinearity and the hard-type anharmonicity DB with frequency above the phonon spectrum. Here we focus on the analysis of the rigidity of the Morse potential on the properties of gap discrete breathers in the 2D biatomic crystal model of composition A3B. Atoms A are ten times heavier than atoms B and this is the reason of the presence of a wide gap in the phonon spectrum of the crystal. The same Morse potentials are used to describe the A-A, B-B, and A-B interactions. Two of the three Morse potential parameters are fixed to unity without the loss of generality. For the third parameter, which defines the stiffness of the interatomic bond, two different values are considered. In both cases two polarizations of DB are investigated. It was found that for a smaller stiffness of the interatomic bond DB have larger maximal amplitude, regardless the type of polarization. On the other hand, they demonstrate a higher rate of small-amplitude waves and thus, they have smaller lifetime.

Propagation of light in photonic lattices with saturable nonlinearity comprising a defect

Linear and nonlinear effects accompanying the propagation of visible laser light through geometrically different one-dimensional photonic lattices made of material with a saturable nonlinearity are investigated, both theoretically and experimentally. Existence, stability and dynamics of various types of nonlinear localized modes (solitons) found to occur in lattices with a defect are explored. Additionally, the phenomenon of spontaneous symmetry breaking of a certain type of solitons is observed in symmetric lattices comprising a defect.

Band gaps and lattice solitons for the higher-order nonlinear Schrödinger equation with a periodic potential

Localization and dynamics of the one-dimensional biharmonic nonlinear Schr\"odinger (NLS) equation in the presence of an external periodic potential is studied. The band-gap structure is determined using the Floquet-Bloch theory and the shape of its dispersion curves as a function of the fourth-order dispersion coupling constant $\ensuremath{\beta}$ is discussed. Contrary to the classical NLS equation ($\ensuremath{\beta}=0$) with an external periodic potential for which a gap in the spectrum opens for any nonzero potential, here it is found that for certain negative $\ensuremath{\beta},$ there exists a nonzero threshold value of potential strength below which there is no gap. For increasing values of potential amplitudes, the shape of the dispersion curves change drastically leading to the formation of localized nonlinear modes that have no counterpart in the classical NLS limit. A higher-order two-band tight-binding model is introduced that captures and intuitively explains most of the numerical results related to the spectral bands. Lattice solitons corresponding to spectral eigenvalues lying in the semi-infinite and first band gaps are constructed. In the anomalous dispersion case, i.e., $\ensuremath{\beta}<0$ (where for the self-focusing nonlinearity no localized nonradiating solitons exist in the absence of an external potential), nonlinear finite-energy stationary modes with eigenvalues residing in the first band gap are found and their properties are discussed. The stability of various localized lattice modes is studied via linear stability analysis and direct numerical simulation.

Strongly localized moving discrete dissipative breather-solitons in Kerr nonlinear media supported by intrinsic gain.

We investigate the mobility of nonlinear localized modes in a generalized discrete Ginzburg-Landau-type model, describing a one-dimensional waveguide array in an active Kerr medium with intrinsic, saturable gain and damping. It is shown that exponentially localized, traveling discrete dissipative breather-solitons may exist as stable attractors supported only by intrinsic properties of the medium, i.e., in the absence of any external field or symmetry-breaking perturbations. Through an interplay by the gain and damping effects, the moving soliton may overcome the Peierls-Nabarro barrier, present in the corresponding conservative system, by self-induced time-periodic oscillations of its power (norm) and energy (Hamiltonian), yielding exponential decays to zero with different rates in the forward and backward directions. In certain parameter windows, bistability appears between fast modes with small oscillations and slower, large-oscillation modes. The velocities and the oscillation periods are typically related by lattice commensurability and exhibit period-doubling bifurcations to chaotically "walking" modes under parameter variations. If the model is augmented by intersite Kerr nonlinearity, thereby reducing the Peierls-Nabarro barrier of the conservative system, the existence regime for moving solitons increases considerably, and a richer scenario appears including Hopf bifurcations to incommensurately moving solutions and phase-locking intervals. Stable moving breathers also survive in the presence of weak disorder.

Concept of a reflective power limiter based on nonlinear localized modes

Optical limiters are designed to transmit low-intensity light, while blocking the light with excessively high intensity. A typical passive limiter absorbs excessive electromagnetic energy, which can cause its overheating and destruction. We propose the concept of a photonic reflective limiter based on resonance transmission via a localized mode. Such a limiter does not absorb the high-level radiation, but rather reflects it back to space. Importantly, the nearly total reflection occurs within a broad frequency range and direction of incidence. The same concept can be applied to infrared and microwave frequencies.

Nonlinear localized modes in [formula omitted]-symmetric optical media with competing gain and loss

The existence and stability of the nonlinear spatial localized modes are investigated in parity-time symmetric optical media characterized by a generic complex hyperbolic refractive index distribution with competing gain and loss profile. The exact analytical expression of the localized modes are found for all values of the competing parameter and in the presence of both the self-focusing and self-defocusing Kerr nonlinearity. The effects of competing gain/loss profile on the stability structure of these localized modes are discussed with the help of linear stability analysis followed by the direct numerical simulation of the governing equation. The spatial localized modes in two-dimensional geometry as well as the transverse power-flow density associated with these localized modes are also examined.

$\mathcal{PT}$ -symmetric nonlinear metamaterials and zero-dimensional systems

A one dimensional, parity-time (${\cal PT}$)-symmetric magnetic metamaterial comprising split-ring resonators having both gain and loss is investigated. In the linear regime, the transition from the exact to the broken ${\cal PT}$-phase is determined through the calculation of the eigenfrequency spectrum for two different configurations; the one with equidistant split-rings and the other with the split-rings forming a binary pattern (${\cal PT}$ dimer chain). The latter system features a two-band, gapped spectrum with its shape determined by the gain/loss coefficient as well as the inter-element coupling. In the presense of nonlinearity, the ${\cal PT}$ dimer chain with balanced gain and loss supports nonlinear localized modes in the form of novel discrete breathers below the lower branch of the linear spectrum. These breathers, that can be excited from a weak applied magnetic field by frequency chirping, can be subsequently driven solely by the gain for very long times. The effect of a small imbalance between gain and loss is also considered. Fundamendal gain-driven breathers occupy both sites of a dimer, while their energy is almost equally partitioned between the two split-rings, the one with gain and the other with loss. We also introduce a model equation for the investigation of classical ${\cal PT}$ symmetry in zero dimensions, realized by a simple harmonic oscillator with matched time-dependent gain and loss that exhibits a transition from oscillatory to diverging motion. This behaviour is similar to a transition from the exact to the broken ${\cal PT}$ phase in higher-dimensional ${\cal PT}-$ symmetric systems. A stability condition relating the parameters of the problem is obtained in the case of piecewise constant gain/loss function that allows for the construction of a phase diagram with alternating stable and unstable regions.

Spontaneous symmetry breaking of gap solitons in defect-loaded uniform one-dimensional photonic lattices

We observe theoretically and experimentally the phenomenon of spontaneous symmetry breaking of strongly localized nonlinear optical modes at the microwatt level in a one-dimensional photonic lattice with a single coupling defect.

Nonlinear localized modes in two-dimensional electrical lattices

We report the observation of spontaneous localization of energy in two spatial dimensions in the context of nonlinear electrical lattices. Both stationary and moving self-localized modes were generated experimentally and theoretically in a family of two-dimensional square as well as honeycomb lattices composed of 6 × 6 elements. Specifically, we find regions in driver voltage and frequency where stationary discrete breathers, also known as intrinsic localized modes (ILMs), exist and are stable due to the interplay of damping and spatially homogeneous driving. By introducing additional capacitors into the unit cell, these lattices can controllably induce mobile discrete breathers. When more than one such ILMs are experimentally generated in the lattice, the interplay of nonlinearity, discreteness, and wave interactions generates a complex dynamics wherein the ILMs attempt to maintain a minimum distance between one another. Numerical simulations show good agreement with experimental results and confirm that these phenomena qualitatively carry over to larger lattice sizes.

Fast pedestal, SOL and divertor measurements from DIII-D to validate BOUT++ nonlinear ELM simulations

This paper documents first work toward validation of BOUT++ nonlinear edge localized mode (ELM) simulations in X-point geometry, at experimental pedestal collisionality, against multiple diagnostic measurements of a well-characterized ELM event in DIII-D. The key to the BOUT++ simulations is the use of a hyper-resistivity model that effectively spreads the very thin current sheets that form in low collisionality nonlinear simulations, and allows for ELM driven magnetic reconnection at finite current density. Experimental ELM characterization includes multiple fast line-integrated diagnostic measurements revealing in–out divertor asymmetric response to ELMs, IRTV imaging at the divertor targets, visible emission in the divertor volume to test the extension of BOUT++ to X-point geometry, and forward modeling of new electron cyclotron emission imaging to test predictions of ELM filaments in the edge pedestal. Initial comparisons suggest optimized BOUT boundary conditions and model parameters, and show similarities between initial BOUT++ results and several measurements.

Nonlinear Wave Packets in Deformed Honeycomb Lattices

The spectrum of a Schrödinger operator with a perfect honeycomb lattice potential has special points, called Dirac points, where the lowest two branches of the spectrum touch. Deformations can result in the merging and disappearance of the Dirac points, and the originally intersecting dispersion relation branches separate. Corresponding to these deformations, nonlinear envelope equations are derived and their dynamics are studied. In the region where Dirac points exist, a maximally balanced equation is derived which has limits to a nonlinear Schrödinger--Kadomtsev--Petviashvili (NLSKP)-type equation and its dispersionless reduction. When the Dirac points disappear and a gap opens, a different maximally balanced equation is derived which has the NLSKP equation and a one-dimensional nonlocal evolution equation as limits. When the gap is sufficiently wide, a nonlinear Dirac equation with nonzero mass and a nonlinear Schrödinger focusing-defocusing system are found. The latter two equations admit nonlinear localized modes. Typical dynamical behaviors of the effective envelope equations are presented.

Breathers in periodic granular chains with multiple band gaps

We consider the localized nonlinear breathing modes that emerge in heterogeneous granular configurations of two materials with a periodicity of three and four beads. We examine as characteristic examples chains with one steel and two aluminum beads, as well as ones with one steel and three aluminum beads, although we briefly touch upon other configurations as well, illustrating their similar characteristics. We analyze the higher order gaps that emerge in such settings and explore the intrinsic localized modes that bifurcate from the edge of the upper bands. A generic surprising feature of such states is that they appear to be more robust than their counterparts bifurcating from the edges of the lower bands. Direct numerical simulations, using driving of the system at suitable frequencies through an actuator or taking advantage of the modulational instabilities of extended band edge states in the system, illustrate the spontaneous formation of localized modes within the corresponding nearest gaps. In these numerical experiments, we also account for the presence of dissipation and analyze its potential role.

Oscillons, solitons and domain walls in arrays of nonlinear plasmonic nanoparticles

The study of metal nanoparticles plays a central role in the emerging novel technologies employing optics beyond the diffraction limit. Combining strong surface plasmon resonances, high intrinsic nonlinearities and deeply subwavelength scales, arrays of metal nanoparticles offer a unique playground to develop novel concepts for light manipulation at the nanoscale. Here we suggest a novel principle to control localized optical energy in chains of nonlinear subwavelength metal nanoparticles based on the fundamental nonlinear phenomenon of modulation instability. In particular, we demonstrate that modulation instability can lead to the formation of long-lived standing and moving nonlinear localized modes of several distinct types such as bright and dark solitons, oscillons, and domain walls. We analyze the properties of these nonlinear localized modes and reveal different scenarios of their dynamics including transformation of one type of mode to another. We believe this work paves a way towards the development of nonlinear nanophotonics circuitry.

Nano breathers and molecular dynamics simulations in hydrogen-bonded chains.

Non-linear localization phenomena in biological lattices have attracted a steadily growing interest and their existence has been predicted in a wide range of physical settings. We investigate the non-linear proton dynamics of a hydrogen-bonded chain in a semi-classical limit using the coherent state method combined with a Holstein-Primakoff bosonic representation. We demonstrate that even a weak inherent discreteness in the hydrogen-bonded (HB) chain may drastically modify the dynamics of the non-linear system, leading to instabilities that have no analog in the continuum limit. We suggest a possible localization mechanism of polarization oscillations of protons in a hydrogen-bonded chain through modulational instability analysis. This mechanism arises due to the neighboring proton-proton interaction and coherent tunneling of protons along hydrogen bonds and/or around heavy atoms. We present a detailed analysis of modulational instability, and highlight the role of the interaction strength of neighboring protons in the process of bioenergy localization. We perform molecular dynamics simulations and demonstrate the existence of nanoscale discrete breather (DB) modes in the hydrogen-bonded chain. These highly localized and long-lived non-linear breather modes may play a functional role in targeted energy transfer in biological systems.

Spontaneous excitation of discrete breathers in crystals with the NaCl structure at elevated temperatures

The density of phonon states at various temperatures has been calculated for crystals with the NaCl structure with equal and strongly differing weights of anions and cations. It has been shown that, in the crystal with a considerable difference in the weights of components, a wide band gap exists in the spectrum of phonon states, which leads to spontaneous excitation of nonlinear localized vibrational modes—gap discrete breathers having frequencies inside the band gap, if the temperature is sufficiently high. It has been found that the peak of the density of phonon states lying above the spectrum of linear vibrations appears at elevated temperatures, which can indicate the existence of discrete breathers with corresponding frequencies. It has been noted that, previously, the existence of gap discrete breathers in the NaI crystal at 555 K was shown experimentally. The presented results bring up the question of the theoretical justification and experimental observation of the breathers with frequencies above the phonon spectrum.

Nonlinear localized modes in Glauber–Fock photonic lattices

We study a nonlinear Glauber-Fock lattice and the conditions for the excitation of localized structures. We investigate the particular linear properties of these lattices, including linear localized modes. We investigate numerically nonlinear modes centered in each site of the lattice. We found a strong disagreement of the general tendency between the stationary and the dynamical excitation thresholds. We define a new parameter that takes into account the stationary and dynamical properties of localized excitations.

Beam propagation in gain–loss balanced waveguides

Linear and nonlinear localized modes of a slab waveguide with distributed gain and loss are studied. The structure is an optical analog of parity-time symmetric potentials in quantum mechanics. Such waveguide structures support stable localized modes. The explicit equation for the propagation constant of linear modes is obtained. The existence of stable nonlinear modes in such waveguides is demonstrated. Bend losses in such structures are analyzed.