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 ut+uux+uxxx−γ1uxxxxx=0,ut+uux+uxxx−γ1uxxxxx=0, and the modified Kawahara–KdV equation ut+3u2ux+uxxx−γ2uxxxxx=0,ut+3u2ux+uxxx−γ2uxxxxx=0, where γi∈Rγi∈R is a positive number when i=1,2i=1,2. The main approach used to determine the stability of solitary traveling waves will be the theory developed by Albert (1992) in [9].