Math 6410 • ODE

Consider the equation
\[
  \frac{\partial u}{\partial t}
  +\frac{\partial}{\partial x}\left(\frac{u^2}{2}\right)
  =\epsilon\frac{\partial^2u}{\partial x^2},
\]
for \(-\infty<x<\infty\) and \(t>0\).
\begin{enumerate}
\item[(a)] Look for a traveling wave solution \(u(x,t)=U(z)\),
\(z=(x-Vt)/\epsilon\), with velocity \(V>0\) and \(U(z)\to U_\pm\) as
\(z\to\pm\infty\). Solve the resulting ODE for \(U(z)\) and deduce that
\[
  V=\frac{[U^2/2]_{-\infty}^{\infty}}{[U]_{-\infty}^{\infty}}
\]
\item[(b)] Discuss how the traveling wave solution relates to shock solutions of the
quasilinear equation obtained by setting \(\epsilon=0\).
\item[(c)] Using phase-plane analysis, show that the wave profile \(U\) is monotonically
decreasing, that is, \(dU/dz<0\).
\end{enumerate}