3,352 research outputs found
Strong transmission and reflection of edge modes in bounded photonic graphene
The propagation of linear and nonlinear edge modes in bounded photonic
honeycomb lattices formed by an array of rapidly varying helical waveguides is
studied. These edge modes are found to exhibit strong transmission (reflection)
around sharp corners when the dispersion relation is topologically nontrivial
(trivial), and can also remain stationary. An asymptotic theory is developed
that establishes the presence (absence) of edge states on all four sides,
including in particular armchair edge states, in the topologically nontrivial
(trivial) case. In the presence of topological protection, nonlinear edge
solitons can persist over very long distances.Comment: 5 pages, 4 figures. Minor updates on the presentation and
interpretation of results. The movies showing transmission and reflection of
linear edge modes are available at
https://www.youtube.com/watch?v=XhaZZlkMadQ and
https://www.youtube.com/watch?v=R8NOw0NvRu
A universal asymptotic regime in the hyperbolic nonlinear Schr\"odinger equation
The appearance of a fundamental long-time asymptotic regime in the two space
one time dimensional hyperbolic nonlinear Schr\"odinger (HNLS) equation is
discussed. Based on analytical and extensive numerical simulations an
approximate self-similar solution is found for a wide range of initial
conditions -- essentially for initial lumps of small to moderate energy. Even
relatively large initial amplitudes, which imply strong nonlinear effects,
eventually lead to local structures resembling those of the self-similar
solution, with appropriate small modifications. These modifications are
important in order to properly capture the behavior of the phase of the
solution. This solution has aspects that suggest it is a universal attractor
emanating from wide ranges of initial data.Comment: 36 pages, 26 pages text + 20 figure
Dispersive shock waves in the Kadomtsev-Petviashvili and Two Dimensional Benjamin-Ono equations
Dispersive shock waves (DSWs) in the Kadomtsev-Petviashvili (KP) equation and
two dimensional Benjamin-Ono (2DBO) equation are considered using parabolic
front initial data. Employing a front tracking type ansatz exactly reduces the
study of DSWs in two space one time (2+1) dimensions to finding DSW solutions
of (1+1) dimensional equations. With this ansatz, the KP and 2DBO equations can
be exactly reduced to cylindrical Korteweg-de Vries (cKdV) and cylindrical
Benjamin-Ono (cBO) equations, respectively. Whitham modulation equations which
describe DSW evolution in the cKdV and cBO equations are derived in general and
Riemann type variables are introduced. DSWs obtained from the numerical
solutions of the corresponding Whitham systems and direct numerical simulations
of the cKdV and cBO equations are compared with excellent agreement obtained.
In turn, DSWs obtained from direct numerical simulations of the KP and 2DBO
equations are compared with the cKdV and cBO equations, again with remarkable
agreement. It is concluded that the (2+1) DSW behavior along parabolic fronts
can be effectively described by the DSW solutions of the reduced (1+1)
dimensional equations.Comment: 25 Pages, 16 Figures. The movies showing dispersive shock wave
propagation in Kadomtsev-Petviashvili II and Two Dimensional Benjamin-Ono
equations are available at https://youtu.be/AExAQHRS_vE and
https://youtu.be/aXUNYKFlke
Two-dimensional localized structures in harmonically forced oscillatory systems
Two-dimensional spatially localized structures in the complex Ginzburg–Landau equation with 1:1 resonance are studied near the simultaneous presence of a steady front between two spatially homogeneous equilibria and a supercritical Turing bifurcation on one of them. The bifurcation structures of steady circular fronts and localized target patterns are computed in the Turing-stable and Turing-unstable regimes. In particular, localized target patterns grow along the solution branch via ring insertion at the core in a process reminiscent of defect-mediated snaking in one spatial dimension. Stability of axisymmetric solutions on these branches with respect to axisymmetric and nonaxisymmetric perturbations is determined, and parameter regimes with stable axisymmetric oscillons are identified. Direct numerical simulations reveal novel depinning dynamics of localized target patterns in the radial direction, and of circular and planar localized hexagonal patterns in the fully two-dimensional system
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
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