The purpose of this course is to examine the properties of various types of soliton-like structures in the vicinity of rest in reversible media. Such media include dispersive media and, for static processes, some dissipative media. Three types of soliton-like structures are considered: classical solitary waves, generalized solitary waves, and solitary wave packets decreasing at spatial infinity. The study of the existence of these soliton-like structures is carried out by projecting an infinite-dimensional system of model equations (partial differential equations) onto the central manifold of the system, followed by a study of the approximation of the resulting dynamic system by integrable equations in normal form. Some properties of the unsteady dynamics of a system with these traveling wave structures are described.

Program:

• The central manifold theorem for partial differential equations.
• Types of bifurcations. The theorem of reduction to a quasi-normal form.
• A simple resonance. Resonance of long and short waves. 1:1-resonance.
• Classic solitary waves. Generalized-solitary waves. Solitary wave packets.
• Plane-parallel movements.
• Long-wave models: capillary and flexural waves. The Kawahara equation.
• Solitary waves in beta plasma. Linear wave resonances, a dynamic system.
• Classic solitary waves. Fast and slow generalized solitary waves.
• Envelope wave packets in the cold plasma region.
• Soliton-like structures in the liquid under the ice cover. Resolventt estimates.
• Spectrum and resonances. Classic solitary waves.
• Generalized-solitary waves.
• Solitary wave packets and dark solitons.

Lecturer
Il'ichev Andrej Teimurazovich

Organizations
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
Steklov International Mathematical Center