Homework Assignment One (January)

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
would be

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!