Algebraic and geometric aspects of soliton equations
The soliton equations are nonlinear wave equations that originated in the 19th century to describe non-decaying waves.
It turned out that some of them are integrable: they have an infinite family of symmetries and can be solved exactly.
The course intends to provide an introduction to the theory of integrable soliton equations.
We will focus on the Korteweg–De Vries and Kadomtsev–Petviashvili equations.
The course will cover basic constructions such as the Lax pair, the tau function, and the Hirota equation.
We will also consider their connection with the representation theory of infinite-dimensional Lie algebras and the theory of Riemann surfaces.
It turned out that some of them are integrable: they have an infinite family of symmetries and can be solved exactly.
The course intends to provide an introduction to the theory of integrable soliton equations.
We will focus on the Korteweg–De Vries and Kadomtsev–Petviashvili equations.
The course will cover basic constructions such as the Lax pair, the tau function, and the Hirota equation.
We will also consider their connection with the representation theory of infinite-dimensional Lie algebras and the theory of Riemann surfaces.
Lecturer
Date
21st March ~ 24th June, 2024
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Monday,Thursday | 10:00 - 11:35 | A3-2-303 | ZOOM 13 | 637 734 0280 | BIMSA |
Prerequisite
Linear algebra, Mathematical analysis, Differential equations. Elementary of Riemann surfases.
Syllabus
- The KdV equation and its symmetries
- Hamilton Formulation
- Quantum inverse scattering method
- Pseudo-differential operators formalism
- Hirota equation and tau-function
- Tau-functions as vacuum expectation values of fermionic operators
- Algebraic geometric of Riemannian surface and integrable models
- Hamilton Formulation
- Quantum inverse scattering method
- Pseudo-differential operators formalism
- Hirota equation and tau-function
- Tau-functions as vacuum expectation values of fermionic operators
- Algebraic geometric of Riemannian surface and integrable models
Reference
- T. Miwa, M. Jimbo, E. Date, "Solitons: Differential Equations, Symmetries and Infinite Dimensional Algebras"
- A. Zabrodin, "Lectures on nonlinear integrable equations and their solutions", https://arxiv.org/pdf/1812.11830.pdf
- A. Alexandrov, A. Zabrodin, "Free fermions and tau-functions", https://arxiv.org/pdf/1212.6049.pdf
- B. A. Dubrovin, "Theta functions and non-linear equations", https://www.mathnet.ru/links/e364e5931b139e7eced69eb579f21659/rm2891_eng.pdf
- B. A. Dubrovin, I. M. Krichever, S. P. Novikov, "Integrable Systems. I", https://homepage.mi-ras.ru/~snovikov/98.pdf
- S. Novikov, S. V. Manakov, L. P. Pitaevskii, V. E. Zakharov, "Theory of Solitons: The Inverse Scattering Method", 1984
- A. Zabrodin, "Lectures on nonlinear integrable equations and their solutions", https://arxiv.org/pdf/1812.11830.pdf
- A. Alexandrov, A. Zabrodin, "Free fermions and tau-functions", https://arxiv.org/pdf/1212.6049.pdf
- B. A. Dubrovin, "Theta functions and non-linear equations", https://www.mathnet.ru/links/e364e5931b139e7eced69eb579f21659/rm2891_eng.pdf
- B. A. Dubrovin, I. M. Krichever, S. P. Novikov, "Integrable Systems. I", https://homepage.mi-ras.ru/~snovikov/98.pdf
- S. Novikov, S. V. Manakov, L. P. Pitaevskii, V. E. Zakharov, "Theory of Solitons: The Inverse Scattering Method", 1984
Audience
Undergraduate
, Advanced Undergraduate
Video Public
Yes
Notes Public
Yes
Language
English
Lecturer Intro
Andrii Liashyk is a researcher in the field of integrated systems, mainly quantum ones. He received his degree from the Center for Advanced Study at Skoltech in 2020. In 2022 he joined BIMSA as a Assistant Professor.