Math 6410 • ODE

Solve (and sketch the solution of) the differential equation \(u_t+uu_x=0\) on the
domain \(-\infty<x<\infty\), \(t>0\) subject to initial data \(u(x,0)=u_0(x)\) where
\begin{enumerate}
\item[(a)]
\begin{equation}
  u_0(x)=
  \begin{cases}
    0, & x<0,\\
    x, & 0<x<1,\\
    1, & x>1,
  \end{cases}
  \tag{2}
\end{equation}
\item[(b)]
\begin{equation}
  u_0(x)=
  \begin{cases}
    1, & x<0,\\
    1-x, & 0<x<1,\\
    0, & x>1.
  \end{cases}
  \tag{3}
\end{equation}
\end{enumerate}
For which (if any) of these initial profiles does the solution of the problem
\(u_t+uu_x=\epsilon u_{xx}\), \(\epsilon>0\), converge to a traveling wave profile?
What is the speed of the traveling wave solution?