Since $\dist$ is a pseudo-metric on $\h{1}(\R)$, $$\dist(\f(t),\f_{c})\leq \dist(\f(t),\f_{c'}) + \dist(\f_{c},\f_{c'})$$ $\dist(\f_{c},\f_{c'})$ is time independent and $$\dist(\f_{c},\f_{c'}) \leq \norm{\f_{c}-\f_{c'}}_\h{1}\leq C \abs{c-c'}\leq C \norm{\f_0-\f_c}_\h{1}$$ which may be made arbitrarily small. So it suffices to study the stability of $\f_{c'}$.

Examining how $\calN[\f]=\calN[\f_c]$ constrains the flow, first decompose $\f$ as $$\f(x+x_0,t) = \f_c(x) + u(x,t)
\label{eq-decomposition}$$ the solitary wave and a perturbation, for some $x_0=x_0(t)$. Expanding $\calN[\f]=\calN[\f_c]$ about $\f_c$, $$\int_{-\infty}^{\infty} \bracket{I - \f_c^n \dx \paren{\f_c^n\dx \cdot} } \paren{\f_c-1} u dx=\inner{ \bracket{I - \f_c^n \dx \paren{\f_c^n\dx \cdot} }\paren{\f_c-1}}{u} = O(\norm{u}_\h{1}^{2})$$ which we make formal in

Given a solitary wave $\f_c$, there exists a constant $C$ such that for any $\norm{u}_\h{1}\leq \frac{1}{2}$, $$\mathcal{N}[\f_c+u]=\mathcal{N}[\f_c]$$ $$\abs{\inner{ \bracket{I - \f_c^n \dx \paren{\f_c^n\dx \cdot}}\paren{\f_c-1}}{u}}\leq C \norm{u}_\h{1}^2
\label{eq:constraint1}$$ \[lem:invariantconstraint\]

Since the right-hand side of is quadratic in $\norm{u}_\h{1}$, we view it as a *near-orthogonality* constraint.

### The Sliding Metric and the Choice of $x_0(t)$ {#sec-sliding-metric}

As discussed in [@Bona75] with regard to the sliding metric, , it is not true in general that the value of $y$ by which one function is translated to minimize the norm is be finite. We will show, under some additional appropriate assumptions that this is the case, see [@Bona75; @Bona94].

Given a solitary 