De l a wa re St a t e Uni ve rs i t y
Department of Applied
Mathematics and Theoretical Physics
Dover, DE 19901
Theory of Solitons
60

8
45

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 Scat
tering 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 t
he 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, Kau
p, Newell and Segur
) scheme; Z
akharov

Shabat 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 a
n 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 Pre

Engineering.
Students with a knowledge in this material can easily proceed to learn advanced m
aterials 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 mate
rial, so as to be able to (1)
follow the applications in
the literature, (2)
solve typical problems in the field, and (3)
discuss a
dequately the term

paper material
.
Topical schedule
:
Basic Topology,
Korteweg

de Vries Equation
Application
:
Fluid Dynamics
I
ntegrals 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 Anal
ysis
Application
:
Fluid Dynamics; Non

linear Optics
C
URRICULUM
C
OURSE
R
EVIEW
:
Theory of Solitons
1.
Course Title/Number:
Theory of Solitons
/
60

8
45

00
2.
Number of Credits:
3
3.
Curriculum Program Title:
Ph.D. in
Applied Mathematics
and Theoretical Physics
4.
Curriculum/Course is:
[ X ]
New
[
]
Revised
[
]
Required Course
[
X
]
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 n
ew 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 involv
e 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; Int
egrals 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 Zakharov

C
URRICULUM
C
OURSE
R
EVIEW
:
Theory of Solitons
Shabat schem
e, Hirota’s bilinear method, Backlund Transform, sine

Gordon equation, Klein

Gordon 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 throug
hout 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 th
e Ph.D. program in Theoretical P
hysics
and Applied
Mathematics
, and is indispensable for students focusing on fundamental physics. For an overview of pre

requisite dependences and the course’s relation to other courses proposed herein, please see the attached
“
Proposed Course Dependencies
” cha
rt.
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 Appl
ied 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.
C
URRICULUM
C
OURSE
R
EVIEW
:
Theory of Solitons
16. How will the course benefit the College?
This course will address ap
plications of ordinary and partial differential equations
in various areas of
fundamental and ap
plied 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 t
o a few select topics in “higher” mathematics and their application in various
branches of physics. This course will be one of the electives speci
fic to the Ph.D. program in Applied
Mathematics and Theoretical Physics of
this department. Besides providing
such a cross

disciplinary 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
3
0 %
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 Wave
s
(Cambridge University Press, 1997; I
SBN = 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 Univers
ity Press ; ISBN = 0

521

38730

2
)
Submitted to Department of Applied Mathematics and Theoret
ical Physics
by: Anjan Biswas
,
on 25th of November
, 2007
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