75 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
Whitham modulation theory for the Kadomtsev-Petviashvili equation
The genus-1 KP-Whitham system is derived for both variants of the
Kadomtsev-Petviashvili (KP) equation (namely, the KPI and KPII equations). The
basic properties of the KP-Whitham system, including symmetries, exact
reductions, and its possible complete integrability, together with the
appropriate generalization of the one-dimensional Riemann problem for the
Korteweg-deVries equation are discussed. Finally, the KP-Whitham system is used
to study the linear stability properties of the genus-1 solutions of the KPI
and KPII equations; it is shown that all genus-1 solutions of KPI are linearly
unstable while all genus-1 solutions of KPII {are linearly stable within the
context of Whitham theory.Comment: Significantly revised versio
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
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