Riemann surfaces and their applications in integrable system

Riemann surfaces and their applications in integrable system

4.88 (4 reviews)
Udemy
platform
English
language
Other Teaching & Academi
category
instructor
1,555
students
2 hours
content
Nov 2020
last update
FREE
regular price

What you will learn

The theory of Riemann surfaces and its applications in integrable models of mathematical physics.

Description

In this course we discuss very interesting and beautiful object - Riemann surfaces. Riemann surfaces have many different applications in integrable systems. And one of our main aim is to explain how Riemann surfaces and their degenerations in singular algebraic curves help to solve problems from geometry and integrable models of mathematical physics. For example, one of such models is a famous Korteweg-de Vries equation:

ut = (6 u uxx +uxxx )/4, u = u(x, t).

This equation describes solitons, that is, solitary water waves in a channel. The theory of Riemann surfaces and its applications in integrable models of mathematical physics.

Sincerely, Andrey Mironov.


Описание курса на русском языке:

В этом курсе мы обсуждаем очень интересные и красивые объекты - римановы поверхности. Римановы поверхности имеют много различных применений в интегрируемых системах. И одна из наших главных целей-объяснить, как римановы поверхности и их вырождения в сингулярных алгебраических кривых помогают решать задачи из геометрии и интегрируемых моделей математической физики. Например, одной из таких моделей является знаменитое уравнение Кортевега-де Фриза:

ut = (6 u uxx +uxxx )/4, u = u(x, t).

Это уравнение описывает солитоны, то есть одиночные волны воды в канале.

Теория римановых поверхностей и ее приложения в интегрируемых моделях математической физики. Мы будем рады видеть Вас на нашем курсе!

Приятного изучения.

С уважением, Андрей Миронов.

Content

Riemann surfaces and their applications in integrable systems

Introduction: Riemann surfaces
Holomorphic function
De finition of a Riemann surface
Examples of Riemann surfaces and Basic theorem of algebra
Complex forms on Rimann surfaces. Part 1
Complex forms on Riemann surfaces. Part 2
Residues of meromorphic 1-forms
Sum of residues. Part 2
Sum of residues. Part 2 (Aplications)
Divisors on Riemann surfaces
Jacobi variety of a compact Riemann surface
Jacobi variety
Kadomtsev-Petviashvilli equation
Abel map
Abel theorem
Applications Riemann surfaces in integrable systems. Korteweg-de Vries equation
Lax representation of Korteweg-de Vries equation
Commuting ordinary difierential operators
Baker-Akhiezer function
Soliton solutions of the Koretweg-de Vries equation

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3623644
udemy ID
11/9/2020
course created date
11/18/2020
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