an analytic function, whose zeros are points in the discrete spectrum of and [*resonances*]{} of the linearized spectral problem about the solitary wave.

Background and Main Results {#sec-preliminaries}
===========================

We first state the basic results on well-posedness of ; see [@Simpson07], Theorems 2.12 and 2.13.

(Local Existence in Time & Continuous Dependence Upon Data)

Let $\phi_{0}$ satisfy $$\norm{\phi_{0}-1}_\h{k}\leq R\quad\mbox{and}\quad \norm{\frac{1}{\phi_{0}}}_{\infty}\leq\frac{1}{2\alpha}$$ for $R>0$, $1\ge \alpha >0$, and $k\ge 1$.

**(a)**

:   There exists $T_\exist>0$, $T_\exist=T_\exist(R,\alpha)$, and $\phi-1\in C^{1}([0,T_\exist):H^{k}(\mathbb{R}))$, such that $\f$ is a solution to with $\phi(\cdot,t)\ge \alpha $ and $\norm{\phi(\cdot,t)-1}_{H^{k}}\leq2 R$ for $t<T_\exist$.

**(b)**

:   There is a maximal time of existence, $T_{\max}>0$, such that if $T_{\max}<\infty$, then $$\label{eq:hkblowup}
                            \lim_{t\to T_{\max}}\norm{\phi(\cdot,t)-1}_{H^{k}}+\norm{\frac{1}{\phi(\cdot,t)}}_{\infty}=\infty$$

**(c)**

:   Let $\phi^{(1)}$, $\phi^{(2)}$ be two solutions of , existing in a common space $C^{1}([0,T):H^{k}(\mathbb{R}))$, $T>0$, and satisfying the bounds $$\norm{\f^{(j)}(t) -1}_\h{k}\leq 2 R \quad\mbox{and}\quad\norm{\frac{1}{\phi^{(j)}(t)}}_{\infty}\leq\frac{1}{\alpha}$$ for some $0< \alpha\leq 1$, $R>0$, $k\geq1$, $j=1,2$, and all $t< T$. Then there exists a constant, $K_\cont=K_\cont(R,\alpha,k)$, such that for any $t$ and $t'$, $0\leq t \leq t' < T$, $$\norm{\f^{(1)}(t') - \f^{(2)}(t')}_\h{k} \leq e^{K_\cont (t'-t)} \norm{\f^{(1)}(t)-\f^{(2)}(t)}_\h{k}
    \label{eq:cont-dep}$$

\[thm:local-existence\]

We will show that the solitary waves of are *orbitally stable* in the following sense. Let us define the distance function,

Let $f-1$ 