see also section \[sec-preliminaries\].

In the article [@Nakayama92], solitary traveling waves, $\f=\f_c(x-ct)$, of speed $c$ are shown to exist for for any speed satisfying $c>n> 1$; $m$ may take any real value. In many problems, solitary waves are well-known to be important coherent structures, participating in key dynamic processes. In order to play this role, solitary waves must be dynamically stable. The most direct approach to the nonlinear dynamic stability of solitary waves is via a variational structure of the equations. Unfortunately, does not appear to have such a structure for the parameter ranges $(m,n)$, arising in the magma problem. However, while not of present interest to the problem of magma we observe that when $n+m=0$, there is a Hamiltonian formulation: $$\begin{aligned}
\label{eq:hamiltonian-form}
\dt\f &=& \bracket{I - \dx(\f^n\dx(\f^{-m}\cdot))}^{-1}\dx (-\f^n) = J \frac{\delta \mathcal{H}}{\delta\phi}\\
\label{eq:hamiltonian}
\mathcal{H}[\f] &=& \int \paren{\frac{1-\f^{n+1}}{n+1}+\f-1}dx\\
\label{eq:skewsym-op}
J=J_\f&=& [I - \dx(\f^n\dx(\f^{-m}\cdot))]^{-1}\dx,\ \ \ n+m=0\end{aligned}$$

We will make use of the Hamiltonian structure in these cases to show that their solitary waves are *orbitally stable*, [*i.e.*]{} for data sufficiently close to a solitary wave, the corresponding solution, modulo a time-dependent spatial translation, will remain close to the solitary wave in $\h{1}(\R)$. The general method of proof is well established 