THESIS
2017
xvii, 118 pages : illustrations (some color) ; 30 cm
Abstract
In this thesis, we study the fast Huygens sweeping method for Schrödinger equations.
We propose a simple backward time marching step to avoid the limitation
on the timestep. We the apply the modication to simulate the multi-color optical self-focusing phenomena in nematic liquid crystals. The propagation of the nematicon is modeled by a parabolic wave equation coupled with a nonlinear
elliptic partial differential equation governing the angle between the crystal and
the direction of propagation. Numerically, the paraxial parabolic wave equation
is solved by a fast Huygens sweeping method, while the nonlinear elliptic PDE
is handled by the alternating direction explicit (ADE) method. The overall algorithm
is shown to be numerically efficient for computing high frequency beam
pro...[
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In this thesis, we study the fast Huygens sweeping method for Schrödinger equations.
We propose a simple backward time marching step to avoid the limitation
on the timestep. We the apply the modication to simulate the multi-color optical self-focusing phenomena in nematic liquid crystals. The propagation of the nematicon is modeled by a parabolic wave equation coupled with a nonlinear
elliptic partial differential equation governing the angle between the crystal and
the direction of propagation. Numerically, the paraxial parabolic wave equation
is solved by a fast Huygens sweeping method, while the nonlinear elliptic PDE
is handled by the alternating direction explicit (ADE) method. The overall algorithm
is shown to be numerically efficient for computing high frequency beam
propagations. Finally, we further extend the method to the nonlocal nonlinear
Schrödinger equation which model the interaction of particles in the Bose-Einstein condensations (BECs).
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