+ u] = \mathcal{E}[\f_c] + \left\langle \frac{\delta \mathcal{E}}{\delta\phi}[\f_c], u \right\rangle+ \frac{1}{2}\left\langle \frac{\delta^2 \mathcal{E}}{\delta\phi^2}[\f_c]u, u \right\rangle + O(\norm{u}_\h{1}^3)$$ The variational derivatives are $$\begin{aligned}
\label{eq:firstvariation}
\frac{\delta \mathcal{E}}{\delta\phi}[\f] &=& c \f -\f^n - c \f^n\dx\paren{\f^n \dx \f} -c+1\\
\label{eq:secondvariation}
L_c u &\equiv \frac{\delta^2 \mathcal{E}}{\delta\phi^2}[\f]u&=-c \dx \paren{{\f}^{2n} \dx u} - \bracket{(2n-1) n c\f^{2n-2}(\dx\f)^2+2 n c{\f}^{2n-1}\dx^2\f +n {\f}^{n-1} -c}u\nonumber \\ \label{eq:lin-op1}
&=& -c\f^n\dx^2\paren{\f^n u}+ \bracket{c -n \f^{n-1} - n c \f^{n-1}\dx\paren{\f^n \dx \f}} u \label{eq:lin-op2}\end{aligned}$$ $\frac{\delta \mathcal{E}}{\delta\phi}[\f_c]=0$ because of ; a solitary wave of speed $c$ is a critical point of the this functional. Alternatively, it can be viewed as critical points of $\mathcal{H}$, subject to the constraint of $\mathcal{N}$ with Lagrange multiplier $c$.

Since the solitary waves are critical points of $\calE$, we would *like* to be able to make an analysis of the form $$\abs{\Delta \mathcal{E}} \geq\frac{1}{2}\inner{ L_c u} {u} \geq C\norm{u}_\h{1}^2 + O\paren{\norm{u}_\h{1}^3},\quad C>0$$ to conclude their Lyapunov stability. However, as proved in Proposition \[prop:Lc\] , the Sturm-Liouville like operator, $L_c$, is not positive definite; it possesses a negative and a zero eigenvalue. Nevertheless, there are natural constraints associated with the problem that will ensure positivity; these will be discussed in Section \[sec:constraints\].

For later use, we state a formal result on

