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 =
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=
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)
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
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 Dire
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 si
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
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
In