Research seminar "Elements of the theory of solites"

Master 2020/2021

Category 'Best Course for New Knowledge and Skills'

Type: Compulsory course (Mathematics)

Area of studies: Mathematics

Delivered by: Department of Fundamental Mathematics

Where: Faculty of Informatics, Mathematics, and Computer Science (HSE Nizhny Novgorod)

When: 1 year, 1-4 module

Mode of studies: offline

Instructors: E.G. Didenkulova, Alexey Slunyaev

Master’s programme: Mathematics

Language: English

ECTS credits: 5

Contact hours: 72

Full Syllabus

Abstract

Solitons – structurally stable wave objects in nonlinear dynamical systems — were discovered in the 1970s in a series of basic nonlinear equations of physics; the class of soliton-like solutions of integrable and non-integrable equations continues to broaden. Solitons are essential or even dominant parts of wave solutions in practically important problems of physics (for example, the data transmission by optical pulses, internal waves and rogue waves in the ocean, non-destructive diagnostics of composite materials) and much beyond. The elaboration of the theory of solitons and of the theory of integrable systems has yielded the appearance and ongoing development of the corresponding mathematical techniques, and also broadening of the field of practical applications. The course will give an introduction to the theory of solitons – as a part of nonlinear dynamics. The distinctive features of these structures among nonlinear waves, the variety of their manifestations (solitons, envelope solitons, breathers) will be shown. The key elements of the Inverse Scattering Technique, and also the basic examples will be discussed. On the basis of multisoliton solutions, the features of the nonlinear wave interaction and focusing will be considered. Actual problems of the scientific research in the fields of nonlinear optics and hydrodynamics will be emphasized.

Learning Objectives

familiarization with solutions of nonlinear partial differential equations of the soliton type, the study of their characteristic properties and typical dynamics, in contrast to linear and nonlinear dispersive waves
formation of the basic knowledge and skills applied to the research of properties of soliton-type waves
awareness of the Inverse Scattering Transform as a nonlinear spectral method for studying wave dynamics
familiarization with actual scientific problems related to the dynamics of soliton-type waves

Expected Learning Outcomes

Aware of the main equations of the wave type, able to recognize nonlinear equations and linear equations with dispersion, can write the dispersion relation for a given linear equation with dispersion, can derive the soliton solution as the localized stationary solution
Aware of the basic properties of the KdV equation, can derive the first integrals of the equation, familiar with the properties of the solitons and their collisions, have a broader view on the KdV-type integrable equations and their soliton-type solutions
Aware of the basic properties of the NLS equation, of the exact solutions bright/gray/dark solutions and breathers, rogue waves, familiar with the concept of the complex amplitude.
Has general understanding of the Inverse Scattering Transform, its similarity and difference from the Fourier method, understands the relation between isolated solutions, discrete eigenspectrum of the associated scattering problem and reflectionless potentials.

Course Contents

Wave equations: linear, nonlinear, with soliton solutions
This Section of the course plays the introductory role. The basic information about waves and wave equations will be given. Brief description of the classic methods of solution of the PDE for waves (Fourier transform, graphic solution of the nonlinear dispersionless equations) will be provided. The first introduction of solitons as stationary solutions of PDEs will be given. Subsections of the course: * Wave equation. Transport equation. Linear equation with dispersion. * Solving linear equations using the Fourier transform. * Nonlinear wave in a medium without dispersion. Gradient catastrophe. * Stationary solutions of wave equations, and solitons.
Korteweg-de Vries equation and its generalizations
This Section of the course is dedicated to the one of most famous integrable equations. By this example the students should be familiarized with the main features which are peculiar for integrable systems. Subsections of the course: * Physical systems leading to the KdV-type equations. * General properties of solutions of the KdV equation. * Properties of solitons of the KdV equation. * Integrals of motions. * Two-soliton solution of the KdV equation. * Solitons and kinks of the modified KdV equation
Nonlinear Schrödinger equation
The concept of modulated waves and complex envelope amplitude are introduced in this Section of the course for the example of the nonlinear Schrodinger equation. The main focus is made on the focusing type of this equation. The general information about the modulational instability and breathers will be given. Subsections of the course: * Physical systems leading to equations of the NLS equation type. * General properties of the NLS equation solutions. * Properties of solitons of the NLS equation of focusing type. * Modulational instability. * Breathers of the NLS equation. Solutions of the rogue-wave type. * Gray and dark solitons of the NLS equation of defocusing type.
Nonlinear spectral analysis
The Inverse Scattering Transform as a nonlinear generalization of the Fourier transform is discussed in the Section of the course with the purpose to give the main idea of the approach and to highlight the common and different sides of these approaches
Generalization and repetition

Assessment Elements

Activity during the seminars
Written examination
The examination time is 120 minutes, performed at the platform et.hse.ru.
Home Works and Short Classworks

Interim Assessment

Interim assessment (2 module)
0.3 * Activity during the seminars + 0.3 * Home Works and Short Classworks + 0.4 * Written examination
Interim assessment (4 module)
0.3 * Activity during the seminars + 0.3 * Home Works and Short Classworks + 0.4 * Written examination

Bibliography

Recommended Core Bibliography

Guo, B., Pang, X.-F., Wang, Y.-F., & Liu, N. (2018). Solitons. Berlin: De Gruyter. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1746464
Shilnikov L.P., Shilnikov A.L., Turaev D.V., Chua L.O. Methods Of Qualitative Theory In Nonlinear Dynamics (Part II). World Sci //Singapore, New Jersey, London, Hong Kong. – 2001.

Recommended Additional Bibliography

Sheng ZHANG, & Caihong YOU. (2019). Inverse Scattering Transform for a Supersymmetric Korteweg-De Vries Equation. Thermal Science, 23, S677–S684. https://doi.org/10.2298/TSCI180512081Z

Course Syllabus