worth noting, that in [@Nakayama91; @Nakayama94], solutions for $c=n$ were found. However, $\f_c(x)=0$ at isolated points and do not fit into our existence theory which relies on boundedness away from zero; hence, we do not consider them here.

also possesses new compacton solutions, discussed in Section \[sec:compacton\].

Our Main Theorem and Corollary \[cor:globalexist\] apply to the Hamiltonian case of , $n+m=0$.

(Orbital Stability)

Let $\f_c$ be a solitary wave with $c>n$ and let $\f-1 \in C^1([0,T); \h{1}(\R))$ be a solution to , $T>0$. There exists $\ep_\star=\ep_\star(\f_c)$ such that for all $\ep\leq \ep_\star$, there is a $\delta$ such that if for some $x_0\in \R$, $$\norm{\f(\cdot, t=0)-\f_c(\cdot-x_0)}_\h{1} < \delta$$ then $\dist(\f(t),\f_c) <\ep$ for all $t \in [0,T)$. \[thm:orbital\]

We make no assumptions about $T$ in this theorem except that $T>0$; indeed it may be infinite.

(Global Existence and Orbital Stability) Given a solitary wave $\f_c$, and $\ep\leq\ep_\star$, there exists $\delta>0$ such that if $$\norm{\f_0-\f_c(\cdot-x_0)}_\h{1}\leq \delta$$ with $x_0\in \R$, then $\f-1 \in  C^{1}([0,\infty):H^{1}(\mathbb{R}))$ and $\dist(\f(t),\f_c) < \ep$ for all time. \[cor:globalexist\]

Conservation Laws and Variational Characterization of Solitary Waves {#sec-invariants-waves}
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Invariants and Regularity
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In addition to the Hamiltonian $\mathcal{H}$, another invariant is the generalized momentum $$\label{eq:momentum}
\mathcal{N}[\f]=\int \paren{\frac{1}{2} \f^{2n}\f_x^2 +\frac{1}{2}(\f-1)^2}dx$$ This 