Southern Federal University
Faculty of mathematics, mechanics and computer science
Milchakova str. 8a, Rostov-on-Don, 344090
Phone.: (863) 2975 111;
Fax: (863) 2975 113;
SYLLABUS
1-semester course
Advanced problems of the mathematical physics
For Masters Program Computational Mechanics and Biomechanics
5 ECTS Credits
Course description
Contemporary approach to science – known as nonlinear science – is more than a new field. Put
simply, it is the recognition that throughout nature, the whole is greater than the sum of parts.
The main purpose of this course is twofold:

To train the students to appreciate the interplay between theory and modeling in problems
arising in the applied science and

To give them a solid theoretical background for nonlinear science
Recommended previous knowledge
Differential equations and the equations of mathematical physics are obligatory. Some background
in continuum mechanics, mathematical models of science is recommended.
Techniques, Skills, etc
After completing the course, the students are expected to be able to:

describe some physics and biomechanical models

develop methods of mathematical physics

analyze the wave propagation

give the proofs of main theoretical results

solve exercises and problems

apply theoretical material to the research work
Requirements and assignments
2 individual tasks, 1 class presentation, 1 final exam, attendance.
Course Content
№
Subject
Assignment
Duration
(in hours)
1.
2.
3.
4.
5.
Introduction.

Soliton. History of discovery

Definition

Modern applications
Conservative Systems with One Degree of Individual Task 1
Freedom

Mathematical Pendulum

Phase plane. Sepatatrix

Exact solutions corresponding the
sepatrix

Elliptic functions

Applications to soliton equations

Presentation
Wave Equations from Mass-Spring Systems

Longitudial motion

Transversal motion

Fourie transform

Laplace transform

Linear waves
Torsion Coupled Pendulums: Sine-Gordon
Equation

Equations for N torsion coupled equal
pendulums

Sine-Gordon equation

Kink and anti-kink solutions

Breather solutions
Backlund Transformation
2
8
8
8
6
6.
7.
8.
9.

Description

Applications to ODE and PDE

Problems
Symmetries of Soliton Equations

Symmetries
groups

KdV equation

The Lax form of an evolution equation

Legendre polynomas

Spherical functions

The quantum mechanics harmonic
oscillator
and
Individual task 2
transformations
Semigroup Theory

Strongly continuous semigroup

A semigroup of contractions
4
The Hirota equation

The Hirota derivative

N-solitons

Vertex operators
8
4
Center Manifolds

Center manifold theorem

Semilinear wave equation

Reduction
10.
The KdV and KP Equations in Physics
11.
Exact Solutions of the KdV and KP equations

The KdV equation and associated
linear system

Some exact solutions of the KdV and
KP equations

Soliton propagation
4
4
Presentation
6
12.
Nerve Pulses and Reactions-Diffusion Systems

Nerve-pulse velocity

Simple nerve models

Reaction
diffusion
dimensions
in
Final exam
6
higher
Grading Policy
Course components
Individual tasks - 20%
Midterm exam - 20%
Class presentation - 20%
Final exam - 25%
Attendance - 15%
Literature
1. Carr J. Applications of centre manifold theory. Springer, 1981. - 160 p.
2. Engel B. J., Nagel L. One-parameter semigroups for linear evolution equations. Springer, 2000. 609 p.
3. Kuznecov Y.A. Elements of applied bifurcation theory. Springer, 1998. - 591 p.
4. Salsa S. Partial Differential Equation in Action – From Modelling to Theory. Springer, 2008.
5. Marcowich P. Applied Partial Differential Equations. Springer, 2007.
6. Miura (ed.) Backlund Transforms, the Inverse Scattering Method, Solitons and Their
Applications. Springer.
7. Miwa (ed.) Solitons: Differential Equations, Symmetries and Infinite Dimensional Algebras.
Cambrige, 2000. - 124 p.
8. Scott A. The Nonlinear Universe. Chaos, Emergence, Life. 2007. 364 p.
9. Scott A. Neuroscience – a Mathematical Primer. Springer, 2002. 378 p.
10. Steeb W. Problems in theoretical physics. Vol. 2. Advanced problems. BI-Wiss.-Verl., 1990. 300 p.
11. Sun J., Luo A. Bifurcation and Chaos in Complex Systems. 2006. 388 p.
General Information
Dr. Svetlana Revina revina@math.sfedu.ru