A note on the stability for Kawahara-KdV type equations

In this paper we establish the nonlinear stability of solitary traveling-wave solutions for the Kawahara-KdV equation $$u_t+u{u}_x+u_{xxx}-{\gamma }_1{u}_{xxxxx}=0,$$ and the modified Kawahara-KdV equation $$u_t+3u^2{u}_x+u_{xxx}-{\gamma }_2{u}_{xxxxx}=0,$$ where ${\gamma }_i\in \mathbb{R}$ is a positive number when $i=1,2$. The main approach used to determine the stability of solitary traveling-waves will be the theory developed by Albert