Section 1
Wave Equations
1.1 Introduction
This rst section of these notes is intended as a very basic introduction to the theory of
wave equations, concentrating on the case of a single space dimension. We will start pretty
much from scratch, not assuming
MATH 337
INTRODUCTION TO SOLITON THEORY
Homework set 5,6
April 18, 2012
Due May 3 , 2012
PROBLEMS.
1. a) Let be an algebra of Laurent series. Let L be given by
ui (x, t) pi
L=
i=
where ui are dierentiable functions of x and t. A bracket cfw_, :
is dened
MATH 337
INTRODUCTION TO SOLITON THEORY
Homework set 3
March 22, 2012
Due April 3 , 2012
PROBLEMS.
1. Find three conservation laws for the mKdV equation
ut 6u2 ux + uxxx
where x R which involve u,u2 , and u4 , respectively.
2. Show that , if u(x, t) is a
SET 8
MATH 543: INTEGRAL REPERSENTATIONS
References: DK and Hildebrandt
Let Lz u(z) = 0 be a homogenous linear dierential equation. Let the integral
representation of the solution of this dierential equation be given by
u(z) =
K(z, t) v(t) dt,
(1)
I
where
SET 7
MATH 543: FUCHSIAN DIFFERENTIAL EQUATIONS
HYPERGEOMETRIC FUNCTION
References: DK and Sadri Hassan.
Historical Notes: Please read the book Linear Dierential Equations and
the Group Theory by Jeremy J. Gray , Birkhouser, 2000 for the contributions
of
MATH. 337
INTRODUCTION TO SOLITON THEORY
May 18 , 2009 Monday 14.40-16.30, SAZ-19
QUESTIONS:
[35]1. a) Let be an algebra of Laurent series. Let L be given by
ui (x, t) pi
L=
i=
where ui are dierentiable functions of x and t. A bracket cfw_, :
is dened a