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

Financial support
The course is supported by the Ministry of Science and Higher Education of the Russian Federation (the grant to the Steklov International Mathematical Center, agreement no. 075-15-2025-303).

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