and discussed in [@Benjamin72; @Bona75; @Weinstein85; @Weinstein86; @Bona94] for the Kortweg - de Vries (KdV), Benjamin - Bona - Mahoney (BBM), and Nonlinear Schrödinger (NLS) equations, amongst others.

We note that in contrast to generalizations of the BBM, KdV, NLS equations to arbitrary power nonlinearity, solitary waves of are nonlinear stable for [*arbitrary*]{} powers, $n$. Currently this is established up to a numerical computation computation of the slope of the function $c\mapsto\calN[\phi_c]$; see Proposition \[prop:increasing-momentum\].

This stability result is also of significance for the global existence theory for ; at present, no global existence in time result is known for the case $n+m=0$. We are required to prove, in tandem with the nonlinear stability, that solutions for data in a neighborhood of a solitary wave exist for all time. Specifically, we note that can potentially become a degenerate dispersive equations if $\f$ tends to zero. As made clear in the well-posedness results [@Simpson07] global existence in time is ensured by uniformly bounding the porosity, $\f$, away from zero. Solitary waves are examples of solutions, whose porosity is uniformly bounded away from zero. The strategy is to show, using spectral and variational arguments, that initial data, in a small $H^1$ neighborhood 