The other two terms follow more easily $$\begin{aligned}
\nonumber\inner{L_c u_\|}{u_\|} &=& \inner{u}{\xi_1}^2\inner{L_c \xi_1}{\xi_1}\\ 
&\geq& -D\norm{u}_\h{1}^{4}
\label{eq:lowerbound2}\end{aligned}$$

$$\begin{aligned}
\nonumber\inner{L_c u_\bot}{u_\|} &=& \inner{L_c u}{ u_\|} - \inner{L_c u_\|}{u_\|} = \inner{u}{\xi_1}\inner{u}{L_c  \xi_1}+\inner{u}{\xi_2}\inner{u}{L_c \xi_2}\\ 
&\geq& -D\norm{u}_\h{1}^{3}
\label{eq:lowerbound3}\end{aligned}$$

then follows immediately from ,, and .

Using the estimate together with , $$\begin{aligned}
\abs{\Delta \calE} &\geq& \calE[\f_c +u]-\calE[\f_c]\geq \frac{1}{2}\inner{L_c u}{u}-D_3 \norm{u}_\h{1}^3\nonumber\\
 &\geq& \frac{1}{2}C_2 \norm{u}_\h{1}^2 - \paren{\frac{1}{2}C_2 +D_3}\norm{u}_\h{1}^3\equiv p_-\paren{\norm{u}_\h{1}}
\label{eq:lower-polynomial}\end{aligned}$$ The polynomial $p_-$ will be important in the next section.

Proof of Orbital Stability and Global Existence in Time
-------------------------------------------------------

Unlike non-degenerate equations, such as KdV and BBM, we need to control $\norm{1/\f}_{L^\infty}$ in our existence proof, hence the additional condition appearing in Theorem \[thm:local-existence\]. It is a lack of *a priori* bounds on this quantity that currently prevents a global existence proof for general $(n,m)$; this matter is discussed in [@Simpson07]. However, we are able to prove, in tandem with nonlinear stability, global existence in time for data in a neighborhood of a solitary wave.

### Proof of Main Theorem {#section:orbital-proof}

Let a particular solitary wave $\f_c$ be given for some $c>n >1$, and let $\f$ be a solution to . First we will consider the case where $\calN[\f]=\calN[\f_c]$; this will then be relaxed.

For $u$ 