Given a Solitary Wave $\f_c$, there exist constants $D_2$ and $D_3$ such that for all such $u\in \h{1}$ with $\norm{u}_\h{1}\leq \frac{1}{2}$. $$\begin{aligned}
\label{eq:energybound1}
\abs{ \calE[\f_c+u]-\calE[\f_c] -\frac{1}{2}\inner{L_c u}{u}}&\leq& D_3\norm{u}_\h{1}^{3}\\
\label{eq:upper-polynomial}
\abs{\calE[\f_c+u]-\calE[\f_c]}& \leq &D_2 \norm{u}_\h{1}^2 + D_3 \norm{u}_\h{1}^3\equiv p_+\paren{\norm{u}_\h{1}}\end{aligned}$$ \[lem:energybound\]

The polynomial $p_+$ will be used in the proof of the main theorem in Section \[section:orbital-proof\].

Another property of the invariant $\calN$ evaluated at $\f_c$ will imply that solitary waves of are never unstable, see Theorem \[thm:orbital\].

(Analytically confirmed for $n=2$, suggested numerically $n\neq2$) \[prop:increasing-momentum\] $$\boxed{\frac{d}{dc}\mathcal{N}[\f_c] >0}
\label{eq-increasing-momentum}$$ for all $c>n>1$.

Integrating again, and applying the boundary condition at $\infty$, $$\label{eq:solitarywave3}
\frac{1}{2}\f_c^{2n} \paren{\dx\f_c}^2 = \frac{1}{2}\paren{\f_c-1}^2 +\frac{1}{c}\bracket{\f_c -1 +\frac{1-\f_c^{n+1}}{n+1}} = F(\f_c;c,n)$$ Using this, and the even symmetry of $\f_c$, $$\calN[\f_c] = 2 \int_0^\infty F(\f_c; c,n) + \frac{1}{2}\paren{\f_c-1}^2 dx$$ can be used to compute $\f_c^{\max}$ by solving $F\f_c^{\max};c,n)=0$. Furthermore, we can compute $dx/d\f_c$ and make a change of variables with it to get $$\label{eq:momentum-solitarywave}
\calN[\f_c] = 2 \int_1^{\f_c^{\max}} \bracket{F(y;c,n) + \frac{1}{2}\paren{y-1}^2}y^n \bracket{2 F(y;c,n)}^{-1/2}dy$$

At present, we have only been able to evaluate analytically in the case $n=2$. Using *Mathematica*, we compute $$\begin{aligned}
\f_c^{\max}&=&\frac{3c -4 }{2}\\
\mathcal{N}[\f_c] &=& \frac{12 \gamma ^3 \left(7 \gamma ^6+93 \gamma ^4+105 \gamma ^2+35\right)}{35 \left(\gamma ^2-1\right)^4} \quad\gamma = \sqrt{\frac{c-2}{c}}\end{aligned}$$ By inspection, 