School of Physics

Lecturer: Dr. Mikhail Kostylev

This is a 6 lecture module for students who have an interest in nonlinear phenomena. We will be mostly interested in dynamical solitons which represent large amplitude waves propagating without spreading and have particle-like properties. However topological solitons like domain walls in magnetic materials will be considered briefly as well. The course is based on advances in nonlinear waves including nonlinear optics, and nonlinear magnetic wave phenomena.

The most important nonlinear differential equations having soliton solutions will be introduced and physical systems described by these equations will be discussed. Experimental evidence for solitary waves in nonlinear media will be presented. We will also discuss applications of solitons in technology.

Course structure

Basic concepts and discovery of solitons.
Korteveg-de Vries Equation (KdV) and its soliton solutions; solitons in shallow water.
Nonlinear Schroedinger Equations (NSE); its occurrence in nonlinear optics, nonlinear magnetic wave phenomena, and Bose-Einstein condensates.
Survey of NSE solutions:

Modulational instability of large amplitude waves and bright envelope solitons in media with attractive nonlinearity;
Dark envelope solitons in media with repulsive nonlinearity;
Real physical systems: effect of weak damping and effects of high-order dispersion and nonlinearity. Applications of optical solitons in optical fibers for communication. Microwave spin wave solitons in ferromagnetic films;
2D NSE: self-focusing, self-channelling and bullet formation in nonlinear optics and spin waves;

Complex Ginzburg-Landau Equation (CGLE) and envelope soliton solutions; generation of trains of envelope soliton-like pulses in optical fiber ring lasers and sequences of microwave pulses in active spin-wave ferromagnetic film based ring resonators;
Sine-Gordon equation and its soliton solutions; fluxons in Josephson transmission lines;
Domain walls as topological solitons.

Solitons

Lecturer: Dr. Mikhail Kostylev

Course structure

Assessment

Take-home exam at the completion of lectures.