2019/2020

Research Seminar "Discrete Integrable Equations and their Reductions"

Category 'Best Course for Career Development'

Category 'Best Course for Broadening Horizons and Diversity of Knowledge and Skills'

Category 'Best Course for New Knowledge and Skills'

Type: Optional course (faculty)

Delivered by: Faculty of Mathematics

Where: Faculty of Mathematics

When: 1, 2 module

Instructors: Andrei Pogrebkov

Language: English

ECTS credits: 3

Contact hours: 30

Course Syllabus

Abstract

Creation and development of the theory of integrable equations is one of main achievements of the mathematical physics of the fall of the previous century. Ideas and results of this theory penetrate in many branches of the modern mathematics: from string theory to the theory of Riemann surfaces. Nowdays essential attention is attracted to the theory of discrete integrable equations. In this lectures a generic approach to construction and investigation of such equations will be presented.

Learning Objectives

• Students will gain an understanding of the main ideas of integrability for difference equations and their consequences.

Expected Learning Outcomes

• Students will be able to construct integrable equations, their Lax pairs
• Students will be able to investigate properties of solutions of integrable difference equations and their reductions

Course Contents

• Commutator identities on associative algebras
• DBAR-problem and dressing operators
• Dressing and Lax pairs
• Hirota difference equation (HDE)
• Higher analogs of HDE
• Direct and the Inverse scattering transform for the HDE
• Soliton solutions
• Two-dimensional reductions, their integrability
• Dispersion relation and integrals of motion
• Other hierarchies of the discrete integrable equations

Assessment Elements

• cumulative grade
  The cumulative grade is proportional to the number of problems solved so that 10 corresponds to 75% of all problems + bonuses for active participants.
• Final exam
  Oral exam. Unless stated otherwise, all grades are rounded to the nearest integer (half-integers are rounded upwards

Interim Assessment

• Interim assessment (2 module)
  0.5 * cumulative grade + 0.5 * Final exam

Bibliography

Recommended Core Bibliography

• Pogrebkov, A. (2016). Commutator identities on associative algebras, the non-Abelian Hirota difference equation and its reductions. Theoretical & Mathematical Physics, 187(3), 823–834. https://doi.org/10.1134/S0040577916060039

Recommended Additional Bibliography

• Hirota Difference Equation and Darboux System: Mutual Symmetry. (2019). Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsbas&AN=edsbas.F587178B