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.
讲师
日期
2024年03月21日 至 06月24日
位置
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| 周一,周四 | 10:00 - 11:35 | A3-2-303 | ZOOM 13 | 637 734 0280 | BIMSA |
修课要求
Linear algebra, Mathematical analysis, Differential equations. Elementary of Riemann surfases.
课程大纲
- 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
参考资料
- 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
听众
Undergraduate
, Advanced Undergraduate
视频公开
公开
笔记公开
公开
语言
英文
讲师介绍
Andrii Liashyk的研究领域是可积系统，主要研究量子系统。他于2020年获得了Skoltech高等研究中心的博士学位。2022年，他加入BIMSA担任助理教授。