and use this as a constraint, as Benjamin did in [@Benjamin72]. This is of course not true for arbitrary $\h{1}$ perturbations. Following, [@Bona75], we show that for $\f$ sufficiently close to $\f_c$, we can find nearby solitary wave $\f_{c'}$ such that $\mathcal{N}[\f] = \mathcal{N}[\f_{c'}] $.

\[lem:nearby-wave\] Given a solitary wave $\f_c$, there exists $\delta_{c}>0$ such that for $\norm{u}_\h{1}\leq \delta_{\calN}$ there exists a $c'$ such that $\mathcal{N}[u+\f_c] = \mathcal{N}[\f_{c'}]$. Furthermore, there exists a constant, $K_c = K_c(\f_c)$ such that $\abs{c-c'}\leq K_c \norm{u}_\h{1}$.

This is proved using the implicit function theorem. Let $\calF$ be the functional $\calF:  \h{1}(\R) \times \R\to \R$ $$\mathcal{F}[u,\Delta c] = \calN[\f_c +u]-\calN[\f_{c+ \Delta c}]$$ $\calF[\mathbf{0},0]=0$ and the Fréchet derivatives are $$D\calF[\bold{0},0]=\twodrowvec{\inner{c^{-1}(\f_c^n-1)}{\cdot}}{-\frac{d}{dc}\mathcal{N}[\f_c]}$$ Both are bounded operators, and by Proposition \[prop:increasing-momentum\] ,$\partial_c\mathcal{N}[\f_c]\neq 0$. We may therefore apply the implicit function theorem for Banach spaces (see [@Reed80] for example), to conclude existence of an $\delta_{\calN}>0$ and a $C^1$ mapping $\mathcal{G}: \h{1} \to \R$, such that if $\norm{u}_\h{1}< \delta_{\calN} $, then $\mathcal{F}[u, \mathcal{G}[u]]=0$. We then set $c' = c +  \mathcal{G}[u]$. Since this map is $C^1$, $\abs{c-c'}\leq K_c \norm{u}_\h{1}$.

If $\f_0$ is our perturbed $\f_c$, and $\norm{\f_0-\f_c}\leq \delta_{c}$, then we may apply Lemma \[lem:nearby-wave\], to find $\f_{c'}$ such that $\calN[\f_0] = \calN[\f_{c'}]$. 