Homework Assignment One (January)

1. Show that an equation of the form

u_{t }= c u_{x }+ e u _{xxx } +
s u u_{x}

can be put in the form

u_{t} = u_{xxx }+ 6 u u_{x}

by a linear change of independent
and dependent variables. Here c, e and s are arbitrary constants.

2. Show that the equation

u_{t }= u_{xxx }+ a u^{(k-1)}u_{x}

has solitons for all constants a when k is an even. integer For which constants a does it have solitons when k is odd? You may use the existence of solitons for k = 2. For KdV (k = 2) the amplitude is proportional to the speed of the soliton. What is the relation for k = 3, 4 , 5 etc ?

3. Look for solutions of the non-linear Schroedinger equation

u_{t }= i(u_{xx} + |u|^{2}u)

of the form u(x,t) =
e^{ibt }f(x+ct). How many solitons are there?
Is the space of solitons a manifold? What is the dimension of this
(putative) manifold?

4. Basic harmonics of a violin or guitar string. (If you haven't done this problem at some time, you really ought to do it now!)

The basic model for small vibrations
of a string under tension of some fixed length is described by the
linear wave equation

m u_{tt }= Tu_{xx}

_{with the boundary conditions
u(0,t) = u(L,t) = 0. Here m is a mass density of the string,
T is proportional to the string tension, and L is the length of the string.
Find all soltions to the equation of the form}

_{
u(x,t) = [a cos wt + b sin wt ] f(x).}

_{Show that the possible frequencies
w which occur are multiples of a fundamental frequency w* , and explain
the relationship of w*to the other constants m,T and L. Does
this fit with your intuition about string instruments? Do you know what
an octave is?}

_{5.* If we
move up to study small vibrations of a circular drum (and we all
know that drums are not as melodious as strings), the equation}
_{would be}

_{ }
m u_{tt }= T(u_{xx }+ u_{yy)}

_{with the boundary condition
that u(x,y) = 0 for x}^{2}_{ + y}^{2}_{
= L}^{2}_{. If we look for solutions
u(x,y,t) =[acos wt + b sin wt]f(x,y), the problem is much harder.
Step one is to write it in polar coordinates, and to separate variables
(again). If you have never heard of a Bessel function, and this interests
you, you might like to look this one up .}_{}

6. Look for solitons of the Sine-Gordon equation
u_{tt }= u_{xx} + sin u. Show that there are two
types of solitons depending on the choice of speed.

_{7. Write out a proof that the
equation y = f(x+F(y)t) defines a function y = u(x,t)
for an interval -T < t < T. To get
the proof to work}
_{you will have to assume that
the function f(x) is bounded? What else will you have to assume about
the functions f and F?}_{}

_{There are seven exercises in
the notes of Palais in section 1.5. Do any number of them!}