Theory and numerics of solitary waves

Abstract

Introduction to solitary waves and solitons in nature, the fundamental PDEs supporting such solutions (including their derivation) and to numerical methods for both fidning these waves as well as for their time evolution.

Objective

Introduction to solitary waves and solitons in nature, the fundamental PDEs supporting such solutions (including their derivation) and to numerical methods for both fidning these waves as well as for their time evolution.

Content

Solitary waves are special solutions to nonlinear PDEs which arise due to a perfect balance between linear dispersive and nonlinear effects. They are localized disturbances that, as the name suggests, evolve without any change to their shape. In cases of completely integrable PDEs they are called solitons. Solitary waves appear in real world as, for instance, laser generated pulses, tidal bores, morning glory clouds, freak waves, tsunami, etc. In this seminar, after briefly covering the history of solitary wave research, we will define a plane wave, phase velocity, wavepacket, group velocity, dispersion relation and the slowly varying envelope approximation. We will next derive some famous soliton carrying PDEs, like the Korteveg de Vries and the Nonlinear Schroedinger equations and sutdy their Hamiltonian structure and the simplest explictly known solitons. We will also concentrate on numerical methods for finding solitary wave solutions in cases when analytic methods fail or are too complicated. The methods include Newton iteration, fixed point iterations, the reduced variational principle and relaxation methods. Another topic in numerics will be the use of split-step and pseudospectral methods for time evolution of the governing PDEs.

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