was identified in [@Simpson07] as a well defined quantity for $\f-1\in\h{1}$, formed out of a linear combination of conservation laws discovered in [@Harris96]. In Appendix \[sec:momentum\], we show the relationship between and the Lagrangian of .

In order to prove that $\mathcal{N}$ and $\mathcal{H}$ are constant in time for $\h{1}$ solutions, one must first establish their conservation in $\h{2}$, which is obvious, and then approximate an $\h{1}$ solution in $\h{2}$ to show that these quantities really are invariant. The time $T_\exist>0$ appearing in Theorem \[thm:local-existence\] is chosen such that both $\mathcal{H}$ and $\mathcal{N}$ are invariant for $\h{1}$ solutions to . See Sections 4 and 5 of [@Simpson07] for details on the invariance of $\mathcal{N}$, which can easily be extended to $\mathcal{H}$.

For $\f-1 \in \h{1}$, $\mathcal{N}$ is obviously well defined; $\mathcal{H}$ is also well defined; the polynomial in the integrand $p(x) = (1-x^{n+1})/(n+1)+x-1$, has $p(1)=p'(1)=0$ and $p''(1)=-n$, giving the bound $$\abs{\mathcal{H[\f]}}\leq C\norm{\f-1}_{L^\infty} \norm{\f-1}_\h{1}^2\leq C \norm{\f-1}_\h{1}^3$$ with $C$ independent of $\f$.

Variational Characterization of the Solitary Waves {#sec:variational}
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Let $$\begin{aligned}
\mathcal{E}[\f] &=& \mathcal{H}[\f] + c \mathcal{N}[\f]\nonumber\\
&=&\int \paren{\frac{1-\f^{n+1}}{n+1} + \f-1}dx + c \int\paren{\frac{1}{2}\f^{2n}\paren{\dx\f}^2 + \frac{1}{2}\paren{\f-1}^2}dx\end{aligned}$$ For $c>n$, consider the taylor expansion of $\calE$ about a solitary wave $\f_c$, $$\label{eq:expansion}
\mathcal{E}[\f_c 