Math 533
Homework 3 Solutions
Remark: This problem set is inspired by a similar one due to Russell
Brown. Let
1
=
+i
2 x
y
1. We compute
+i
(u + iv )
x
y
= (ux vy ) + i (vx + uy )
=0
2g =
so that ux = vy and uy = vx . If u, v C 2 (R2 ) we may dierentiate
Math 773
Problem Set #2
Solutions
1. Consider the Dirac -distribution, dened for a 2 Rd by
a
['] = '(a):
(a) We compute the Fourier transform using the denition:
b [' ] =
a
[' ]
b
= '(a)
b
a
= (2 )
d=2
Z
e
ia y
'(y ) dy
and read o from this identity that
Math 773
Problem Set #3
Solutions
To avoid depleting the nation supply of parentheses, we denote by Lq Lp
s
tx
the spaces Lq (Lp ) in what follows. Aso, note that h( ) is a polynomial but
t
x
h(x; t) is a function on Rd R!
1. For both directions we will u
Math 773
Problem Set #1
Solutions
1. (Johnson, Q3.1) Show that a scaling transformation u ! u, x ! x,
t ! t for nonzero real constants , , and , transforms the general
KdV equation
Aut + Buux + Cuxxx = 0
into
ut
6uux + uxxx = 0:
(Remark: The form above is
LECTURE 3: COMPLETELY INTEGRABLE EQUATIONS I: THE
KORTEWEG-DE VRIES EQUATION
PETER PERRY
Contents
1. Introduction
2. Direct and Inverse Scattering for the Schrdinger Equation on the Line
2.1. The Direct Scattering Map
2.2. The Inverse Scattering Map
3. Ti
INVERSE SCATTERING FOR THE DEFOCUSSING
DAVEY-STEWARTSON II EQUATION
PETER PERRY
Abstract. We rework the treatment of DS II in Perry previous paper, using
s
the Lax representation to give direct and simple proofs that the inverse scattering method produces
LECTURE 1: WATER WAVE EQUATIONS, LONG WAVES, AND
WEAKLY NONLINEAR WAVES
PETER PERRY
Contents
1. Water Wave Equations
1.1. Free Surface Problem
1.2. Dimensionless Equations for the Free Surface Problem
1.3. Boundary Conditions for the Rescaled Free Surface
LECTURE 2: FOURIER ANALYSIS AND DISPERSIVE LINEAR
EQUATIONS
PETER PERRY
Contents
1. Introduction
2. Review of Fourier Analysis
3. Interpolation, Young Inequality, and the Hardy-Littlewood-Sobolev
s
Inequalities
4. Solving Linear Evolution Equations
4.1. T