1. Show that an equation of the form
ut = c ux + e u xxx + s u ux
can be put in the form
ut = uxxx + 6 u ux
by a linear change of independent
and dependent variables. Here c, e and s are arbitrary constants.
2. Show that the equation
ut = uxxx + a u(k-1)ux
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
ut = i(uxx + |u|2u)
of the form u(x,t) = eibt 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 utt = Tuxx
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
m utt = T(uxx + uyy)
with the boundary condition that u(x,y) = 0 for x2 + y2 = L2. 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 utt = uxx + 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!