of the solitary wave, remain in a small neighborhood, therefore persists in being bounded away from zero, ensuring global existence and stability.

We proceed as follows. In Section \[sec-preliminaries\], we review some of the basic mathematical properties of on well-posedness theory and solitary waves, and state the our main results: Theorem \[thm:orbital\] and Corollary \[cor:globalexist\]. In Section \[sec-invariants-waves\], we review the constants of motion and their relation to the solitary waves. The proofs of the main results, Theorem \[thm:orbital\] and Corollary \[cor:globalexist\], on orbital stability and global existence of data near a solitary wave solution, are given Section \[sec-stability-proof\]. In Section \[sec:compacton\], we note a relationship between our equations and those that have *compacton* solutions, solitary waves with compact support, [@Rosenau06a], and show that also possesses such solutions.

Finally, we remark that in a forthcoming paper, we prove the *asymptotic* stability of solitary waves in the general case (arbitrary $m$ and $n$) of , of small amplitude, without any restriction on $m$ and $n$ . In fact, the Hamiltonian structure which we presently use in the case $n+m=0$ for has implications for the linear spectral theory and stability analysis in this work, via the Evans function (see, for example, [@Pego92]), 