scholarly journals Designing Optical Fibers: Fitting the Derivatives of a Nonlinear Pde-Eigenvalue Problem

2012 ◽  
Vol 02 (04) ◽  
pp. 321-330 ◽  
Author(s):  
Linda Kaufman ◽  
Seok-Min Bang ◽  
Brian Heacook ◽  
William Landon ◽  
Daniel Savacool ◽  
...  
Keyword(s):  
Eigenvalue Problem ◽  
Optical Fibers ◽  
Nonlinear Pde ◽  
Derivatives Of

Lump solutions to nonlinear PDEs involving Hirota derivative Dt2DxDy

2020 ◽  
Vol 34 (18) ◽  
pp. 2050197
Author(s):  
Fudong Wang ◽  
Wen Xiu Ma
Keyword(s):  
Nonlinear Pde ◽  
Nonlinear Pdes ◽  
Bilinear Method ◽  
Lump Solutions ◽  
New Class ◽  
Derivative Term ◽  
Pde Systems ◽  
Derivatives Of ◽  
Hirota Bilinear ◽  
Temporal Variable

This paper aims to study lump solutions to a class of (2[Formula: see text]+[Formula: see text]1)-dimensional nonlinear PDE systems, which involve the fourth-order Hirota derivative term: [Formula: see text]. This Hirota derivative term generates higher-order derivatives of the temporal variable. Lump solutions to the resulting new class of nonlinear PDE systems are studied in detail via the Hirota bilinear method.

An Efficient Calculation of Photonic Crystal Band Structures Using Taylor Expansions

2014 ◽  
Vol 16 (5) ◽  
pp. 1355-1388 ◽  
Author(s):  
Dirk Klindworth ◽  
Kersten Schmidt
Keyword(s):  
Photonic Crystal ◽  
Eigenvalue Problem ◽  
Brillouin Zone ◽  
Taylor Expansion ◽  
Dispersion Curves ◽  
Band Structures ◽  
One Dimensional ◽  
The One ◽  
Derivatives Of ◽  
Selection Of

AbstractIn this paper we present an efficient algorithm for the calculation of photonic crystal band structures and band structures of photonic crystal waveguides. Our method relies on the fact that the dispersion curves of the band structure are smooth functions of the quasi-momentum in the one-dimensional Brillouin zone. We show the derivation and computation of the group velocity, the group velocity dispersion, and any higher derivative of the dispersion curves. These derivatives are then employed in a Taylor expansion of the dispersion curves. We control the error of the Taylor expansion with the help of a residual estimate and introduce an adaptive scheme for the selection of nodes in the one-dimensional Brillouin zone at which we solve the underlying eigenvalue problem and compute the derivatives of the dispersion curves. These derivatives are then employed in a Taylor expansion of the dispersion curves. We control the error of the Taylor expansion with the help of a residual estimate and introduce an adaptive scheme for the selection of nodes in the one-dimensional Brillouin zone at which we solve the underlying eigenvalue problem and compute the derivatives of the dispersion curves. The proposed algorithm is not only advantageous as it decreases the computational effort to compute the band structure but also because it allows for the identification of crossings and anti-crossings of dispersion curves, respectively. This identification is not possible with the standard approach of solving the underlying eigenvalue problem at a discrete set of values of the quasi-momentum without taking the mode parity into account.

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