Delaware State University
Department of Applied Mathematics and Theoretical Physics
Dover, DE 19901
Theory of Solitons
60-845-00,
3 cr.
Text:
P. G. Drazin & R. S. Johnson: Solitons: An Introduction
recommended:
R. S. Johnson:
A Modern Introduction to the Mathematical Theory of Water Waves
M. J. Ablowitz & H. Segur: Solitons and the Inverse Scattering Transform
M. J. Ablowitz & P. A. Clarkson: Solitons, Nonlinear Evolution Equations and Inverse Scattering
The aim of the course is to introduce the basic concepts of the mathematical aspects of Soliton Theory.
This will include the derivation and the introduction to the Korteweg-de Vries equation; the traveling
wave solution, Inverse Scattering Transform; N-soliton solution; Lax pair; Integrals of Motion; Hirota’s
bilinear method; Backlund Transform; AKNS (Ablowitz, Kaup, Newell and Segur) scheme; ZakharovShabat scheme; Painleve transcedents; Painleve conjecture; perturbation of solitons; adiabatic parameter
dynamics; Topological solitons, kinks and anti-kinks, breathers, phonons, skyrimions; chiral solitons.
The aim of this course is to serve as an introductory course to the more advanced course entitled Optical
Solitons that is offered in the Ph.D. program in Optics from the Department of Physics and PreEngineering. Students with a knowledge in this material can easily proceed to learn advanced materials
in optical solitons or topological solitons and other areas of Theoretical Physics.
Prerequisite: Advanced Calculus and Partial Differential Equations or equivalent. A successful student is
expected to gain a working knowledge of the covered material, so as to be able to (1) follow the
applications in the literature, (2) solve typical problems in the field, and (3) discuss adequately the termpaper material.
Topical schedule:





Basic Topology, Korteweg-de Vries Equation
Application: Fluid Dynamics
Integrals of Motion
Application: Fluid Dynamics and Electromagnetic Theory
Inverse Scattering Transform
Application: Quantum Mechanics, Non-linear Optics
Backlund Transform
Application: Partial Differential Equations and Integrability Issues
Painleve Analysis
Application: Fluid Dynamics; Non-linear Optics
CURRICULUM COURSE REVIEW:
Theory of Solitons
1. Course Title/Number:
Theory of Solitons / 60-845-00
2. Number of Credits:
3
3. Curriculum Program Title:
Ph.D. in Applied Mathematics and Theoretical Physics
4. Curriculum/Course is:
[X]
New
[
[
Required Course
[X]
]
]
Revised
Elective Course
5. List Prerequisites:
25-561/562 (Real Analysis), or equivalent
60-853 (Partial Differential Equations)
6. List Courses Being Replaced or Changed:
This is a new course.
7. List Courses Being Deleted:
No courses are being deleted.
8. Needs Statement:
This course is needed for students pursuing a Ph.D. in all areas of theoretical physics and Applied
mathematics. The frameworks of Shallow Water Waves, optical pulses, topological solitons involve the
basic structure and properties of topological and non-topological solitons. The course material will cover
the following topics:
Korteweg-de Vries equation; the traveling wave solution, Inverse Scattering Transform; N-soliton
solution; Lax pair; Integrals of Motion; Hirota’s bilinear method; Backlund Transform; AKNS scheme;
Zakharov-Shabat scheme; Painleve transcedents; Painleve conjecture; perturbation of solitons; adiabatic
parameter dynamics; Topological solitons, kinks and anti-kinks, breathers, phonons, skyrimions; Chiral
solitons.
This course gives an unified description of all the topics that are necessary to cover these advanced
materials. This course lays the foundations to understanding these phenomena in fundamental and
applied physics.
9. Catalog Description of the Course:
This course introduces the concept of topological and non-topological solitons. The emphasis will be on
mathematical structure and properties that includes the inverse scattering transform, AKNS and
CURRICULUM COURSE REVIEW:
Theory of Solitons
Zakharov-Shabat scheme, Hirota’s bilinear method, Backlund Transform, sine-Gordon equation, KleinGordon equation, sinh-Gordon equation, Painleve analysis, soliton perturbation theory.
10. List of Objectives of the Course:
(1) To provide an introduction to the body knowledge and techniques of integrability studies
of nonlinear evolution equations.
(2) To see how these techniques apply to the analysis of phenomena in Fluid Dynamics, Plasma Physics,
and Theoretical Physics.
(3) To learn how to identify those phenomena throughout theoretical physics and Applied Mathematics.
(4) To develop the problem-solving skills associated with the application of these methods in theoretical
physics, and learn how to extract experimentally verifiable information from such application.
11. Course Outline:
See the “Topical schedule” section in the attached brief syllabus.
12. Show how the proposed course fits into the curriculum or course sequence:
This course is an elective within the curriculum of the Ph.D. program in Theoretical Physics and Applied
Mathematics, and is indispensable for students focusing on fundamental physics. For an overview of prerequisite dependences and the course’s relation to other courses proposed herein, please see the attached
“Proposed Course Dependencies” chart.
13. Are there comparable courses in other departments?
No.
14. How will the students be affected by this course change?
This course provides the students an opportunity to increase their integration with the research program
of the Department of Applied Mathematics and Theoretical Physics, by understanding the mathematical
underpinnings of the techniques that are used in contemporary theoretical physics. This course will
improve students’ professional competence, employability in technical fields and ability to pass
professional examinations. Neither this course nor its prerequisites increase the total number of semester
hours in this curriculum or the number of credit hours required for graduation.
15. What effect will this new course have on College resource?
None: this course will not require new or additional resources or staffing.
CURRICULUM COURSE REVIEW:
Theory of Solitons
16. How will the course benefit the College?
This course will address applications of ordinary and partial differential equations in various
areas of fundamental and applied physics, some of which lie at the foundation of numerous other
disciplines in science: engineering (e.g., optical fibers and lasers), physics (e.g. Nonlinear Optics).
17. How will the change affect the program?
This course will introduce students to a few select topics in “higher” mathematics and their application
in various branches of physics. This course will be one of the electives specific to the Ph.D. program in
Applied Mathematics and Theoretical Physics of this department. Besides providing such a crossdisciplinary broadening of knowledge for the students in this program, it also serves as a prerequisite to
Optical Solitons, also proposed herein.
18. Evaluation of Student Performance:
Homework Assignments
15 %
Two (2) in-term examinations
30 %
Term-paper
15 %
Final Examination
40 %
Sample homework assignments, in-term and final examination question-sheets, work sheets,
course notes, review sheets and term papers will be accessible on-line.
Course Structure: Three (3) 50-minute lectures per week.
References
1. P. G. Drazin & R. S. Johnson: Solitons: An Introduction
(Cambridge University Press, 1992; ISBN = 0-521-33655-4)
2. R. S. Johnson: A Modern Introduction to the Mathematical Theory of Water Waves
(Cambridge University Press, 1997; ISBN = 0-521-59832-X)
3. M. J. Ablowitz & H. Segur: Solitons and Inverse Scattering Transform
(SIAM Publishers, 1981; ISBN = 0-89871-174-6)
4. M. J. Ablowitz & P. A. Clarkson: Solitons, Nonlinear Evolution Equations and Inverse
Scattering (Cambridge University Press ; ISBN = 0-521-38730-2)
Submitted to Department of Applied Mathematics and Theoretical Physics
by: Anjan Biswas, on 25th of November, 2007