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### MATH 337 - INTRODUCTION TO SOLITON THEORY

### Fall 2002, Spring 2009, Spring 2012, Spring 2014

**Math337 **: Introduction to Soliton Theory. An Undergraduate
elective Course.

** **__Text book: __
** Solitons ** by P.G. Drazin and R.S. Johnson ,
Cambridge University. London Mathematical
Society Lecture Note Series ** 85 **.

** **__ Course Schedule: __
Wednesday 13.40-15.30 SA-Z01
Friday 15.40-17.30 SAZ-Z01

__ Exams: __
(%25) First Midterm Exam (2009)
, (2012) : March 21, 2014
(%25) Second Midterm Exam (2009),
(2012) : May 2, 2014
(%26) Final Exam (2009) ,
(2012) :xx,2014
(%24) Homework:

Homework assignments

** FZ Grade Policy:**
Failure to attend at least 50% of the classes or averaging less than 40%
from the two midterms will result in an FZ grade.

__ Lectures __

Lecture 1

Lecture 2

Lecture 3

Lecture 4

Lecture 5

**Subjects Covered **

1. ** 6 February **

Dispersion, Dissipiation and nonlinearity
Solitary waves and Solitons
The KdV equation and its solitary waves

Lecture 1

First Homework set

2. ** 13 February **

Symmetries of the KdV equation
Reduction of KdV to ODEs (Group Invariant Solutions)
Travelling wave solutions of KdV
Solutions in terms of Jacobi Elliptic functions
Cnoidal waves , solitary waves

3. ** 20 February **

Group invariant solutions of Burgers Equation
Group invariant solutions of modified Korteweg-de Vries Equation
Group invariant solutuins of Nonlinear Schrodinger Equation

4. ** 27 February **

The scattering problem
Sturm-Liouville eigenvalue problem
Delta function and sech^2 function
(We omitted the section 3.3)

Second Homework set

Lecture 2

5. ** 5 March **
Lax Equations
Time evolution of the scattering data
Solutions of the GLM equation
One and two soliton solutions of the KdV equation

6. ** 12 March **
Lax Formulation
The KdV Hierarchy
Inverse scattering of the KdV Hierarchy

Lecture 3

7. ** 19 March **

First Midterm Exam
Solutions of the first midterm exam
One soliton solution of the KdV hierarchy

Third Homework set

8. ** 26 March **
The Hirota method (for the KdV equation)

9. ** 2 April **
Backlund transformations
Backlund transformations (for the KdV equation)

Fourth Homework set

10. ** 9 April **
Other Backlund transformations
Nonlinear superposition rule (for KdV and SG equations)

Lecture 4

11. ** 16 April **

Fifth Homework set

Variational Calculus

12. ** 23 April **

Hamiltonian formulation of KdV equation

13. ** 30 April **

AKNS Formulation

Lecture 5

Second Midterm Exam(2012)

14. ** 07 May **

Poisson Brackets

Hydrodynamic Type of Integrable Equations

** **__Course Syllabus__

1. Sept.23. The Koteweg de Vries Eqiuation

2. Sept.30. The Korteweg de Vries Equation

3. Oct.07. The Cnoidal waves

4. Oct.14. The Cnoidal waves.

5. Oct.21. The Conservation Laws

6. Oct.28. The Conservation Laws

7. Oct.25. The Initial Value problem of the KdV equation

8. Nov.01. The Initial Value problem of the KdV equation

9. Nov.08. The Lax Method

10. Nov.15. The Lax Method

11. Nov.22. Sine Gordon equation

12. Nov.29. Sine Gordon Equation

13. Dec.06. Backlund Transformations

14. Dec.13. Backlund Transformations.

15. Dec.20. AKNS System

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### Last update January 2014

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