AM107 AM107  Takehome Exam (Problem 3)
Handed out: Thursday, November 30, 2000
Return by:  5:00pm on Thursday, December 14, 2000

Problem 3: Consider the model

ut = -uv
vt = uv + vxx
where the subscripts represent derivatives.  We will search for traveling waves in these equations.

a. Moving to a traveling frame, let U(z) = u(x,t) and V(z) = v(x,t) where z = x - ct.  Write down the two higher-order ODE's corresponding to the PDE's above written in this new frame.
 

b.  Usually, we would introduce a new variable W = Vz and write the equations in a. as a system of three first order equations in U, V and W.  Instead,  add the equation for U to the equation for V and note the cancellation of the UV-terms.
 

c.  Integrate this equation once to get a single first order equation for Vz.  (Don't forget the constant of integration....)
 

d.  Now we have a two-dimensional system.  Find the steady states and evaluate their stability (as a function of a and c) to determine whether or not there may exist a trajectory connecting the two steady states.
 

e. Use phase-plane techniques (or pplane5) to verify the existence of this connecting orbit for appropriate values of the parameters.  Plot U and V vs. z for this trajectory to exhibit the qualitative shape of this traveling wave (pplane5 can do this too).
 

f.  Hypothesize about what sort of system these equations might model.