this is strictly increasing for $c>2$. The general behavior, both in this case and the rest, is diagrammed in Figure \[fig-momentum-cartoon\].

For $n\neq 2$, we justify our result with a computer plot, shown in Figure \[fig-functional-manifold\]. The manifold, $\mathcal{M}=(n,c,\mathcal{N}[\f_c])$, is monotonically increasing in $c$ for fixed $n$. This was computed by first solving $F(\f_c^{\max};c,n)=0$ for $\f_c^{\max}$ using Brent’s Method. was then integrated using the QUADPACK routine QAWS to handle the $(\phi_c^{\max}-y)^{-1/2}$ singularity. We used the GNU Scientific Library (GSL) implementation of these two methods, [@Galassi06].

![A diagram of $\calN[\f_c]$. $\calN[\f_c]$ and $d/dc \calN[\f_c]$ are zero at $c=n$. For $c\leq n$, there are not solitary waves.[]{data-label="fig-momentum-cartoon"}](momentum-cartoon){width="3.5in"}

![A plot of $\log\mathcal{N}[\f_c]$, as a function of both $c$ and $n$, $c>n>1$, for the set $(c,n) \in [1.5,9.5]\times[1.5,9.5]$. It is monotonic in both arguments, and in particular, increasing in $c$ for fixed $n$.[]{data-label="fig-functional-manifold"}](log-manifold-diagram-bw){width="5in"}

Orbital Stability and Global Existence {#sec-stability-proof}
======================================

Constraints of the Flow {#sec:constraints}
-----------------------

Under appropriate restrictions on our function space, $L_c$ will be a positive definite operator, allowing us to conclude orbital stability. This is accomplished via two constraints discussed in the following two sections, \[sec:momentum-invariant\] and \[sec-sliding-metric\].

### The $\mathcal{N}$ Invariant {#sec:momentum-invariant}

We would like to assume $\mathcal{N}[\f] = \mathcal{N}[\f_c]$, 