# Delaware State University

Mechanics

Oct 24, 2013 (4 years and 6 months ago)

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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
-

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