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1. Standing and traveling waves in a minimal nonlinearly dispersive lattice model

   Parker, R; Germain, P; Cuevas_Maraver, J; Aceves, A; Kevrekidis, PG (June 2024, Physica D Nonlinear phenomena)

   In the work of Colliander et al. (2020) a minimal lattice model was constructed describing the transfer of energy to high frequencies in the defocusing nonlinear Schrödinger equation. In the present work, we present a systematic study of the coherent structures, both standing and traveling, that arise in the context of this model. We find that the nonlinearly dispersive nature of the model is responsible for standing waves in the form of discrete compactons. On the other hand, analysis of the dynamical features of the simplest nontrivial variant of the model, namely the dimer case, yields both solutions where the intensity is trapped in a single site and solutions where the intensity moves between the two sites, which suggests the possibility of moving excitations in larger lattices. Such excitations are also suggested by the dynamical evolution associated with modulational instability. Our numerical computations confirm this expectation, and we systematically construct such traveling states as exact solutions in lattices of varying size, as well as explore their stability. A remarkable feature of these traveling lattice waves is that they are of ‘‘antidark’’ type, i.e., they are mounted on top of a non-vanishing background. These studies shed light on the existence, stability and dynamics of such standing and traveling states in 1 + 1 dimensions, and pave the way for exploration of corresponding configurations in higher dimensions. 

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2. Standing and traveling waves in a model of periodically modulated one-dimensional waveguide arrays

   Parker, Ross; Aceves, Alejandro; Cuevas_Maraver, Jesus; Kevrekidis, PG (August 2023, Physical review E)

   In the present work, we study coherent structures in a one-dimensional discrete nonlinear Schrodinger lattice in which the coupling between waveguides is periodically modulated. Numerical experiments with single-site initial conditions show that, depending on the power, the system exhibits two fundamentally different behaviors. At low power, initial conditions with intensity concentrated in a single site give rise to transport, with the energy moving unidirectionally along the lattice, whereas high power initial conditions yield stationary solutions. We explain these two behaviors, as well as the nature of the transition between the two regimes, by analyzing a simpler model where the couplings between waveguides are given by step functions. For the original model, we numerically construct both stationary and moving coherent structures, which are solutions reproducing themselves exactly after an integer multiple of the coupling period. For the stationary solutions, which are true periodic orbits, we use Floquet analysis to determine the parameter regime for which they are spectrally stable. Typically, the traveling solutions are characterized by having small-amplitude, oscillatory tails, although we identify a set of parameters for which these tails disappear. These parameters turn out to be independent of the lattice size, and our simulations suggest that for these parameters, numerically exact traveling solutions are stable. 

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3. Traveling-Standing Water Waves

   https://doi.org/10.3390/fluids6050187  

   Wilkening, Jon (May 2021, Fluids)

   We propose a new two-parameter family of hybrid traveling-standing (TS) water waves in infinite depth that evolve to a spatial translation of their initial condition at a later time. We use the square root of the energy as an amplitude parameter and introduce a traveling parameter that naturally interpolates between pure traveling waves moving in either direction and pure standing waves in one of four natural phase configurations. The problem is formulated as a two-point boundary value problem and a quasi-periodic torus representation is presented that exhibits TS-waves as nonlinear superpositions of counter-propagating traveling waves. We use an overdetermined shooting method to compute nearly 50,000 TS-wave solutions and explore their properties. Examples of waves that periodically form sharp crests with high curvature or dimpled crests with negative curvature are presented. We find that pure traveling waves maximize the magnitude of the horizontal momentum among TS-waves of a given energy. Numerical evidence suggests that the two-parameter family of TS-waves contains many gaps and disconnections where solutions with the given parameters do not exist. Some of these gaps are shown to persist to zero-amplitude in a fourth-order perturbation expansion of the solutions in powers of the amplitude parameter. Analytic formulas for the coefficients of this perturbation expansion are identified using Chebyshev interpolation of solutions computed in quadruple-precision. 

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4. Lasing Action from Quasi‐Propagating Modes

   https://doi.org/10.1002/adma.202203999  

   Tan, Max_J_H; Park, Jeong‐Eun; Freire‐Fernández, Francisco; Guan, Jun; Juarez, Xitlali_G; Odom, Teri_W (July 2022, Advanced Materials)

   Abstract Band edges at the high symmetry points in reciprocal space of periodic structures hold special interest in materials engineering for their high density of states. In optical metamaterials, standing waves found at these points have facilitated lasing, bound‐states‐in‐the‐continuum, and Bose–Einstein condensation. However, because high symmetry points by definition are localized, properties associated with them are limited to specific energies and wavevectors. Conversely, quasi‐propagating modes along the high symmetry directions are predicted to enable similar phenomena over a continuum of energies and wavevectors. Here, quasi‐propagating modes in 2D nanoparticle lattices are shown to support lasing action over a continuous range of wavelengths and symmetry‐determined directions from a single device. Using lead halide perovskite nanocrystal films as gain materials, lasing is achieved from waveguide‐surface lattice resonance (W‐SLR) modes that can be decomposed into propagating waves along high symmetry directions, and standing waves in the orthogonal direction that provide optical feedback. The characteristics of the lasing beams are analyzed using an analytical 3D model that describes diffracted light in 2D lattices. Demonstrations of lasing across different wavelengths and lattice designs highlight how quasi‐propagating modes offer possibilities to engineer chromatic multibeam emission important in hyperspectral 3D sensing, high‐bandwidth Li‐Fi communication, and laser projection displays. 

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5. Nonreciprocal Pattern Formation of Conserved Fields

   https://doi.org/10.1103/PhysRevX.14.021014  

   Brauns, Fridtjof; Marchetti, M Cristina (April 2024, Physical Review X)

   In recent years, non-reciprocally coupled systems have received growing attention. Previous work has shown that the interplay of non-reciprocal coupling and Goldstone modes can drive the emergence of temporal order such as traveling waves. We show that these phenomena are generically found in a broad class of pattern-forming systems, including mass-conserving reaction--diffusion systems and viscoelastic active gels. All these systems share a characteristic dispersion relation that acquires a non-zero imaginary part at the edge of the band of unstable modes and exhibit a regime of propagating structures (traveling wave bands or droplets). We show that models for these systems can be mapped to a common ``normal form'' that can be seen as a spatially extended generalization of the FitzHugh--Nagumo model, providing a unifying dynamical-systems perspective. We show that the minimal non-reciprocal Cahn--Hilliard (NRCH) equations exhibit a surprisingly rich set of behaviors, including interrupted coarsening of traveling waves without selection of a preferred wavelength and transversal undulations of wave fronts in two dimensions. We show that the emergence of traveling waves and their speed are precisely predicted from the local dispersion relation at interfaces far away from the homogeneous steady state. The traveling waves are therefore a consequence of spatially localized coalescence of hydrodynamic modes arising from mass conservation and translational invariance of displacement fields. Our work thus generalizes previously studied non-reciprocal phase transitions and identifies generic mechanisms for the emergence of dynamical patterns of conserved fields. 

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