arXiv subjects

Justin T. Cole

Publications and source records attributed to Justin T. Cole.

At least 19 recordsLinked to original sources

arXiv2026

Chiral solitary waves in a nonlinear topological insulator model

An outstanding challenge in the field of topological insulators is the realization of nonlinear systems that support coherent traveling waves. Highly nonlinear lattices can suffer from significant radiation losses due to Peierls-Nabarro effects. In this work a nonlinear tight-binding model that supports robust traveling edge states is proposed and examined. This system possess a nontrivial local Chern topology and soliton-like states. When a traveling solitary wave collides with a stationary mode, the two are observed to interact inelastically. These results suggest future directions for the modeling, realization, and application of nonlinear Chern insulators.

nlin.PS↗

arXiv2026

Topological routing in Chern insulators

Chern insulator systems are realizable in numerous physical systems and can support robust nonreciprocal transmission of energy. A routing functionality constructed from two counter-oriented Chern insulator regions, using coupled Haldane type systems is proposed. By adjusting the strength of a magnetic field and the frequency of an antenna source, it possible to steer the flow of energy: completely to the left, completely to the right, or split. Alternatively, two sources can be used to direct the flow of energy. This formulation has the potential to serve as a robust and reconfigurable component in optical transmission.

physics.optics↗

arXiv2026

A split-step Christov method for approximating rational PDE solutions

Rational solutions of partial differential equations (PDEs) are notoriously difficult to approximate via spectral Fourier methods due to their algebraically slow decay rate. In this work we discuss approximating rational PDE solutions in a basis of orthogonal functions known as the Fourier series, allowing for the computation of its spectrum via the fast Fourier transform. Spectral differentiation matrices are derived. Several explicit fourth-order split-step integrators are derived and their performance compared. As an application, rogue wave solutions in a family of nonlinear Schrödinger equations are explored. Perturbing the constant background is found to generate rogue wave-like structures. The effects of higher-order dispersion and generalized nonlinearities are also examined.

nlin.PS↗

arXiv2025

Spiral Wave Solutions in Water Waves

Spiral wave solutions are found in linear and weakly nonlinear irrotational water wave equations. These unsteady spiral waves evolve from suitable initial conditions; they are not induced by external forcing. In the linear case, a long-time asymptotic result is obtained via the method of stationary phase. The asymptotic approximation is found to be in good agreement with the exact solution and reveals hyperbolic spiral structure. Numerical simulations show that these spiral waves persist in the presence of weak nonlinearity. While spiral solutions are frequently found in excitable media governed by reaction-diffusion systems, they comprise a new class of interesting two space one time dimensional solutions in fundamental linear and nonlinear dispersive wave systems.

physics.flu-dyn↗

arXiv2024

Quantitative measure of topological protection in Floquet systems through the spectral localizer

The standard understanding of topological protection from band theory is that a system's topology cannot change without first closing the bulk band gap. However, in Floquet systems, this typical definition of topological protection is one step removed from the experimentally accessible system parameters, as the relationship between the disorder in a system's instantaneous Hamiltonian and its Floquet Hamiltonian that defines its topology is not straightforward. Here, we demonstrate that the spectral localizer framework for classifying material topology can be applied to Floquet systems and prove that its associated measure of topological protection can be understood in terms of the integrated disorder across the system's instantaneous Hamiltonians. As such, we have derived a quantitative bound on a Floquet system's topological protection in terms of the instantaneous system. Moreover, we show the utility of these bounds in both ordinary an anomalous Floquet Chern insulators.

physics.optics↗

arXiv2024

Data-driven approximations of topological insulator systems

A data-driven approach to calculating tight-binding models for discrete coupled-mode systems is presented. Specifically, spectral and topological data is used to build an appropriate discrete model that accurately replicates these properties. This work is motivated by topological insulator systems that are often described by tight-binding models. The problem is formulated as the minimization of an appropriate residual (objective) function. Given bulk spectral data and a topological index (e.g. winding number), an appropriate discrete model is obtained to arbitrary precision. A nonlinear least squares method is used to determine the coefficients. The effectiveness of the scheme is highlighted against a Schrödinger equation with a periodic potential that can be described by the Su-Schrieffer-Heeger model.

cond-mat.mes-hall↗

arXiv2024

Collapse dynamics for two-dimensional space-time nonlocal nonlinear Schrodinger equations

The question of collapse (blow-up) in finite time is investigated for the two-dimensional (non-integrable) space-time nonlocal nonlinear Schrodinger equations. Starting from the two-dimensional extension of the well known AKNS q,r system, three different cases are considered: (i) partial and full parity-time (PT) symmetric, (ii) reverse-time (RT) symmetric, and (iii) general q,r system. Through extensive numerical experiments, it is shown that collapse of Gaussian initial conditions depends on the value of its quasi-power. The collapse dynamics (or lack thereof) strongly depends on whether the nonlocality is in space or time. A so-called quasi-variance identity is derived and its relationship to blow-up is discussed. Numerical simulations reveal that this quantity reaching zero in finite time does not (in general) guarantee collapse. An alternative approach to the study of wave collapse is presented via the study of transverse instability of line soliton solutions. In particular, the linear stability problem for perturbed solitons is formulated for the nonlocal RT and PT symmetric nonlinear Schrodinger (NLS) equations. Through a combination of numerical and analytical approaches, the stability spectrum for some stationary one soliton solutions is found. Direct numerical simulations agree with the linear stability analysis which predicts filamentation and subsequent blow-up.

nlin.PS↗

arXiv2023

Switching via Wave Interaction in Topological Photonic Lattices

A honeycomb Floquet lattice with helically rotating waveguides and an interface separating two counter-propagating subdomains is analyzed. Two topologically protected localized waves propagate unidirectionally along the interface. Switching can occur when these interface modes reach the edge of the lattice and the light splits into waves traveling in two opposite directions. The incoming mode, traveling along the interface, can be routed entirely or partially along either lattice edge with the switching direction based on a suitable mixing of the interface modes.

physics.optics↗

arXiv2022

Nonlinear optical waveguide lattices: Asymptotic analysis, solitons, and topological insulators

In recent years, there has been considerable interest in the study of wave propagation in nonlinear photonic lattices. The interplay between nonlinearity and periodicity has led researchers to manipulate light and discover new and interesting phenomena such as new classes of localized modes, usually referred to as solitons, and novel surface states that propagate robustly. A field where both nonlinearity and periodicity arises naturally is nonlinear optics. But there are other areas where waves propagating on background lattices play an important role, including photonic crystal fibers and Bose-Einstein condensation. In this review article the propagation of wave envelopes in one and two-dimensional periodic lattices associated with additional potential in the nonlinear Schrodinger (NLS) equation, termed lattice NLS equations, are studied. A discrete reduction, known as the tight-binding approximation, is employed to find the linear dispersion relation and the equations governing nonlinear discrete envelopes for two-dimensional simple periodic lattices and two-dimensional non-simple honeycomb lattices. In the limit under which the envelopes vary slowly, continuous envelope equations are derived from the discrete system. The coefficients of the linear evolution system are related to the dispersion relation in both the discrete and continuous cases. For simple lattices, the continuous systems are NLS type equations. In honeycomb lattices, in certain cases, the continuous system is found to be nonlinear Dirac equations. Finally, it is possible to realize so-called topological insulator systems in an optical waveguide setting. The modes supported by these systems are associated with spectral topological invariants and, remarkably, can propagate without backscatter from lattice defects.

nlin.PS↗

arXiv2022

Soliton-mean field interaction in Korteweg-de Vries dispersive hydrodynamics

The propagation of localized solitons in the presence of large-scale waves is a fundamental problem, both physically and mathematically, with applications in fluid dynamics, nonlinear optics and condensed matter physics. Here, the evolution of a soliton as it interacts with a rarefaction wave or a dispersive shock wave, examples of slowly varying and rapidly oscillating dispersive mean fields, for the Korteweg-de Vries equation is studied. Step boundary conditions give rise to either a rarefaction wave (step up) or a dispersive shock wave (step down). When a soliton interacts with one of these mean fields, it can either transmit through (tunnel) or become embedded (trapped) inside, depending on its initial amplitude and position. A comprehensive review of three separate analytical approaches is undertaken to describe these interactions. First, a basic soliton perturbation theory is introduced that is found to capture the solution dynamics for soliton-rarefaction wave interaction in the small dispersion limit. Next, multiphase Whitham modulation theory and its finite-gap description are used to describe soliton-rarefaction wave and soliton-dispersive shock wave interactions. Lastly, a spectral description and an exact solution of the initial value problem is obtained through the Inverse Scattering Transform. For transmitted solitons, far-field asymptotics reveal the soliton phase shift through either type of wave mentioned above. In the trapped case, there is no proper eigenvalue in the spectral description, implying that the evolution does not involve a proper soliton solution. These approaches are consistent, agree with direct numerical simulation, and accurately describe different aspects of solitary wave-mean field interaction

nlin.PS↗

arXiv2022

Unified approach to Floquet lattices, topological insulators, and their nonlinear dynamics

A unified method to analyze the dynamics and topological structure associated with a class of Floquet topological insulators is presented. The method is applied to a system that describes the propagation of electromagnetic waves through the bulk of a two-dimensional lattice that is helically-driven in the direction of propagation. Tight-binding approximations are employed to derive reduced dynamical systems. Further asymptotic approximations, valid in the high-frequency driving regime, yield a time-averaged system which governs the leading order behavior of the wave. From this follows an analytic calculation of the Berry connection, curvature and Chern number by analyzing the local behavior of the eigenfunctions near the critical points of the spectrum. Examples include honeycomb, Lieb and kagome lattices. In the nonlinear regime novel equations governing slowly varying wave envelopes are derived. For the honeycomb lattice, numerical simulations show that for relatively small nonlinear effects a striking spiral patterns occurs; as nonlinearity increases localized structures emerge and for somewhat higher nonlinearity the waves collapse.

nlin.PS↗

arXiv2021

On the Whitham system for the (2+1)-dimensional nonlinear Schrödinger equation

We derive the Whitham modulation equations for the nonlinear Schrödinger equation in the plane (2d NLS) with small dispersion. The modulation equations are derived in terms of both physical and Riemann variables; the latter yields equations of hydrodynamic type. The complete 2d NLS Whitham system consists of six dynamical equations in evolutionary form and two constraints. As an application, we determine the linear stability of one-dimensional traveling waves. In both the elliptic and hyperbolic case, the traveling waves are found to be unstable. This result is consistent with all previous investigations of such stability by other methods. These results are supported by direct numerical calculations.

nlin.PS↗

arXiv2021

Transverse Instability of Rogue Waves

Rogue waves are abnormally large waves which appear unexpectedly and have attracted considerable attention, particularly in recent years. The one space, one time (1+1) nonlinear Schrödinger equation is often used to model rogue waves; it is an envelope description of plane waves and admits the so-called Pergerine and Kuznetov-Ma soliton solutions. However, in deep water waves and certain electromagnetic systems where there are two significant transverse dimensions, the 2+1 hyperbolic nonlinear Schrodinger equation is the appropriate wave envelope description. Here we show that these rogue wave solutions suffer from strong transverse instability at long and short frequencies. Moreover, the stability of the Peregrine soliton is found to coincide with that of the background plane wave. These results indicate that, when applicable, transverse dimensions must be taken into account when investigating rogue wave pheneomena.

nlin.PS↗

arXiv2021

Peierls-Nabarro barrier effect in nonlinear Floquet topological insulators

The Peierls-Nabarro barrier is a discrete effect that frequently occurs in discrete nonlinear systems. A signature of the barrier is the slowing and eventual stopping of discrete solitary waves. This work examines intense electromagnetic waves propagating through a periodic honeycomb lattice of helically-driven waveguides, which serves as a paradigmatic Floquet topological insulator. Here it is shown that discrete topologically protected edge modes do not suffer from the typical slowdown associated with the Peierls-Nabarro barrier. Instead, as a result of their topological nature, the modes always move forward and redistribute their energy: a narrow (discrete) mode transforms into a wide effectively continuous mode. On the other hand, a discrete edge mode that is not topologically protected does eventually slow down and stop propagating. Topological modes that are initially narrow modes naturally tend to wide envelope states that are described by a generalized nonlinear Schrodinger equation. These results provide insight into the nature of nonlinear topological insulators and their application.

nlin.PS↗

arXiv2020

Whitham equations and phase shifts for the Korteweg-deVries equation

The semi-classical Korteweg-deVries equation for step-like data is considered with a small parameter in front of the highest derivative. Using perturbation analysis Whitham theory is constructed to higher order. This allows the order one phase and the complete leading order solution to be obtained; the results are confirmed by extensive numerical calculations.

nlin.PS↗

arXiv2020

Discrete Approximation of Topologically Protected Modes in Magneto-Optical Media

Topologically protected waves in the linearly polarized Maxwell's equations with gyrotropic, magneto-optic media were studied a decade ago both computationally and experimentally. This paper develops a robust tight-binding model for this system that makes careful use of Wannier function representations. The model provides very good approximations to the underlying band structure. When solved on a semi-infinite strip, it produces exponentially localized edge modes whose corresponding eigenvalues span the frequency band gaps. A set of coupled differential equations are derived which allows one to find how the electromagnetic field propagates unidirectionally, without backscatter from defects. Furthermore, the discrete model predicts topologically protected edge modes with nontrivial Chern number which are consistent with direct simulation.

physics.optics↗

arXiv2019

Topological insulators in longitudinally driven waveguides: Lieb and Kagome lattices

Topological insulators are studied via tight-binding approximations of longitudinally driven photonic lattices with three lattice sites per unit cell. Two cases are considered in detail: Lieb and Kagome lattices. The lattice is decomposed into three sublattices each of which are allowed move independently of one another. Emphasis is placed on periodic driving induced by laser-etched helical coils along the direction of propagation. The linear Floquet bands are constructed for various inter-sublattice rotation patterns such as: different radii, different frequency, phase offset and quasi one-dimensional motion. Depending on the nature of the band structure, bulk spectral bands with nonzero Chern number are found to support topologically protected edge states which can move unidirectionally. In this case, the modes move scatter-free around defects due to underlying topological protection. Intriguing mode dynamics are found including bi-directional topological modes and bulk-edge leakage i.e. excitation of bulk modes at a defect for edge modes with dispersion frequencies nearby the bulk bands. Finally, certain nonlinear edge modes are also found to propagate unidirectionally and scatter-free around lattice defects.

nlin.PS↗

arXiv2018

On the Whitham system for the radial nonlinear Schrödinger equation

Dispersive shock waves (DSWs) of the defocusing radial nonlinear Schrödinger (rNLS) equation in two spatial dimensions are studied. This equation arises naturally in Bose-Einstein condensates, water waves and nonlinear optics. A unified nonlinear WKB approach, equally applicable to integrable or nonintegrable partial differential equations, is used to find the rNLS Whitham modulation equation system in both physical and hydrodynamic type variables. The description of DSWs obtained via Whitham theory is compared with direct rNLS numerics; the results demonstrate very good quantitative agreement. On the other hand, as expected, comparison with the corresponding DSW solutions of the one-dimensional NLS equation exhibits significant qualitative and quantitative differences.

nlin.PS↗