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\begin{center}
UNM Dept. of Mathematics and Statistics \\
Math. 570, Asymptotics and Singular Perturbations \\
Problem Set III \\[.2in]
\end{center}
Spring 2011 \\[.2in]
\begin{enumerate}

\item Consider the integral 
\[ f(t) = \int_0^\infty exp\left( t ( z + i z - z^3) \right)dz \ .\]
Use the saddle point method to show that, for $t \rightarrow \infty$
this integral is aprroximated by
\[ f(t) 
\thicksim\!\!\!\!\!\!^{{}^{\textrm{asy}}}
e^{-i\pi/16}2^{-1/8}3^{-1/4}\pi^{1/2}t^{-1/2}
                exp\left\{2^{7/4} 3^{-3/2}e^{3i\pi/8}t\right\}
\ .\] 

(Hint: use the saddle point on the first quadrant).

\item Asymptotics for the Klein-Gordon equation
\[ \phi_{tt} - \phi_{xx} + \phi = 0\]
\begin{enumerate}
\item Show that the dispersion relation is
\[ w(k) = \left( 1+k^2\right)^{1/2} \ .\]
\item An initially localized disturbance $\phi(x,0) = f(x)$, 
$\phi_t(x,0) = 0$, is followed by an observer along a ray moving at
speed $x/t = v >0$ away from the source. Give the asymptotic form of the
disturbance for large $t$.
\end{enumerate}

\item Consider a chain of unit masses, coupled by springs obeying potential
$U(v_i,i\in {\bf{Z}})$ where $v_i$ is the longitudal displacement of
the $i-$th mass from its position of equilibrium at location $x_i = i \Delta x$.
Here, $U$ is the Fermi-Pasta-Ulam potential
\[ U = \sum_{i=-\infty}^\infty \frac{\beta}{4} \left(v_{i+1}-v_i\right)^4
+\frac{1}{2} \left(v_{i+1}-v_i\right)^2 \ .\]
Show that if one considers displacements that have a slow variation from one
position to the next then in the 'continuous' limit and in the frame moving with the group velocity to the right
one obtains the PDE:
\[ u_t + A u^2 u_x + B u_{xxx} = 0, \ \
\]
where $u\propto v_x = \sfrac{v_{i+1}-v_{i-1}}{2\Delta x}$.

Find the coefficients A, B in terms of the 'nonlinearity' parameter $\beta$.

Find a scaling of $x$, $t$ and $u$ that leads to the canonical form
\[  u_t + u^2 u_x + u_{xxx} = 0 \ .\
\]

\item This problem examines the difference between dissipation and dispersion.
Consider the PDE
\begin{eqnarray}
\phi_t + (1 + \epsilon \phi)\phi_x + \mu \phi_{xxx} & = & 0 \label{KdV} \\
u_t + (1 + \epsilon u)u_x + \nu u_{xx} & = & 0 \label{Burgers}
\end{eqnarray}
These are "small nonlinearity" versions of the {\it Korteweg-de Vries}
({\bf KdV}, \ref{KdV}) and {\it Burgers} (\ref{Burgers}) equations. Show that the
dispersion relations of the linearized ($\epsilon = 0$) equations are, 
respectively
\begin{eqnarray*}
w(k) & = &  k -  \mu k^3 \\
w(k) & = &  k + i\nu k^2  \ .
\end{eqnarray*}
In each case, derive the large t asymptotic form for a small, 
localized perturbation followed along a ray $x/t = v > 0$.


\end{enumerate}
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