Orbital stability and dynamical behaviors of solitary waves for the Camassa–Holm equation with quartic nonlinearity

Stability of Negative Solitary Waves for a Generalized Camassa-Holm Equation with Quartic Nonlinearity

“…In this paper, we consider this stability problem. Unlike the results for positive solitary waves in [39], we show that there exist negative solitary waves for any wave speed > 0. The point lying in our results is that we can actually determine that the scalar function ( ) (see below) is convex with respect to wave speed ; that is, all the negative solitary waves are orbitally stable.…”

“…Our study is closely related to the results in [39]. For convenience, we write (2) when = 3 in the following form:…”

“…For = 3 and = 2, the negative solitary wave to (2) was obtained and proved to be orbitally stable for any speed in [22]. Moreover, the stability problem of solitary wave to (2) was investigated as = 3 and > 0 in [39].…”

“…In [39], when the parameter > 0, (3) was shown to be Painlevé nonintegrable and to have positive solitary waves as wave speed > √10/ . The solitary waves were proved to be unstable when the wave speed tends to the critical value √10/ and stable while the wave speed is a little bigger than the critical value.…”