arXiv subjects

M. J. Ablowitz

Publications and source records attributed to M. J. Ablowitz.

9 recordsLinked to original sources

arXiv2023

The Haldane Model in a Magneto-optical Honeycomb Lattice

A two-dimensional honeycomb lattice composed of gyrotropic rods is studied. Beginning with Maxwell's equations, a perturbed Wannier method is introduced which yields a tight-binding model with nearest and next-nearest neighbors. The resulting discrete model leads to a Haldane model and as such, topologically protected modes, associated with nonzero Chern numbers are supported. Changing the radii of the rods allows for the breaking of inversion symmetry which can change the topology of the system. This model describes experimental results associated with topological waves in magneto-optical honeycomb lattices. This method can also be applied to more general Chern insulator lattices. When on-site Kerr type nonlinear effects are considered, coherent soliton-like modes are found to propagate robustly through boundary defects.

physics.optics↗

arXiv2015

Conical Wave Propagation and Diffraction in 2D Hexagonally Packed Granular Lattices

Linear and nonlinear mechanisms for conical wave propagation in two-dimensional lattices are explored in the realm of phononic crystals. As a prototypical example, a statically compressed granular lattice of spherical particles arranged in a hexagonal packing configuration is analyzed. Upon identifying the dispersion relation of the underlying linear problem, the resulting diffraction properties are considered. Analysis both via a heuristic argument for the linear propagation of a wavepacket, as well as via asymptotic analysis leading to the derivation of a Dirac system suggests the occurrence of conical diffraction. This analysis is valid for strong precompression i.e., near the linear regime. For weak precompression, conical wave propagation is still possible, but the resulting expanding circular wave front is of a non-oscillatory nature, resulting from the complex interplay between the discreteness, nonlinearity and geometry of the packing. The transition between these two types of propagation is explored.

nlin.PS↗

arXiv2014

Adiabatic Dynamics of Edge Waves in Photonic Graphene

The propagation of localized edge modes in photonic honeycomb lattices, formed from an array of adiabatically varying periodic helical waveguides, is considered. Asymptotic analysis leads to an explicit description of the underlying dynamics. Depending on parameters, edge states can exist over an entire period or only part of a period; in the latter case an edge mode can effectively disintegrate and scatter into the bulk. In the presence of nonlinearity, a `time'-dependent one-dimensional nonlinear Schrödinger (NLS) equation describes the envelope dynamics of edge modes. When the average of the `time varying' coefficients yields a focusing NLS equation, soliton propagation is exhibited. For both linear and nonlinear systems, certain long lived traveling modes with minimal backscattering are found; they exhibit properties of topologically protected states.

physics.optics↗

arXiv2009

Soliton Generation and Multiple Phases in Dispersive Shock and Rarefaction Wave Interaction

Interactions of dispersive shock (DSWs) and rarefaction waves (RWs) associated with the Korteweg-de Vries equation are shown to exhibit multiphase dynamics and isolated solitons. There are six canonical cases: one is the interaction of two DSWs which exhibit a transient two-phase solution, but evolve to a single phase DSW for large time; two tend to a DSW with either a small amplitude wave train or a finite number of solitons, which can be determined analytically; two tend to a RW with either a small wave train or a finite number of solitons; finally, one tends to a pure RW.

nlin.PS↗

arXiv2007

The piston dispersive shock wave problem

The one-dimensional piston shock problem is a classical result of shock wave theory. In this work, the analogous dispersive shock wave (DSW) problem for a dispersive fluid described by the nonlinear Schrödinger equation is analyzed. Asymptotic solutions are calculated using Whitham averaging theory for a "piston" (step potential) moving with uniform speed into a dispersive fluid at rest. These asymptotic results agree quantitatively with numerical simulations. It is shown that the behavior of these solutions is quite different from their classical counterparts. In particular, the shock structure depends on the speed of the piston. These results have direct application to Bose-Einstein condensates and the propagation of light through a nonlinear, defocusing medium.

nlin.PS↗

arXiv2006

On Dispersive and Classical Shock Waves in Bose-Einstein Condensates and Gas Dynamics

A Bose-Einstein condensate (BEC) is a quantum fluid that gives rise to interesting shock wave nonlinear dynamics. Experiments depict a BEC that exhibits behavior similar to that of a shock wave in a compressible gas, eg. traveling fronts with steep gradients. However, the governing Gross-Pitaevskii (GP) equation that describes the mean field of a BEC admits no dissipation hence classical dissipative shock solutions do not explain the phenomena. Instead, wave dynamics with small dispersion is considered and it is shown that this provides a mechanism for the generation of a dispersive shock wave (DSW). Computations with the GP equation are compared to experiment with excellent agreement. A comparison between a canonical 1D dissipative and dispersive shock problem shows significant differences in shock structure and shock front speed. Numerical results associated with the three dimensional experiment show that three and two dimensional approximations are in excellent agreement and one dimensional approximations are in good qualitative agreement. Using one dimensional DSW theory it is argued that the experimentally observed blast waves may be viewed as dispersive shock waves.

cond-mat.other↗

arXiv2005

Theory of Magnetodynamics Induced by Spin Torque in Perpendicularly Magnetized Thin Films

A nonlinear model of spin wave excitation using a point contact in a thin ferromagnetic film is introduced. Large-amplitude magnetic solitary waves are computed, which help explain recent spin-torque experiments. Numerical simulations of the fully nonlinear model predict excitation frequencies in excess of 0.2 THz for contact diameters smaller than 6 nm. Simulations also predict a saturation and red shift of the frequency at currents large enough to invert the magnetization under the point contact. The theory is approximated by a cubic complex Ginzburg-Landau type equation. The mode's nonlinear frequency shift is found by use of perturbation techniques, whose results agree with those of direct numerical simulations.

cond-mat.other↗

arXiv1998

On Integrability and Chaos in Discrete Systems

The scalar nonlinear Schrodinger (NLS) equation and a suitable discretization are well known integrable systems which exhibit the phenomena of ``effective'' chaos. Vector generalizations of both the continuous and discrete system are discussed. Some attention is directed upon the issue of the integrability of a discrete version of the vector NLS equation.

solv-int↗

arXiv1998

On Discretizations of the Vector Nonlinear Schrodinger Equation

Two discretizations of the vector nonlinear Schrodinger (NLS) equation are studied. One of these discretizations, referred to as the symmetric system, is a natural vector extension of the scalar integrable discrete NLS equation. The other discretization, referred to as the asymmetric system, has an associated linear scattering pair. General formulae for soliton solutions of the asymmetric system are presented. Formulae for a constrained class of solutions of the symmetric system may be obtained. Numerical studies support the hypothesis that the symmetric system has general soliton solutions.

solv-int↗