Orbital stability of solitary waves of compound KdV-type equation

In this paper, we study the orbital stability of solitary waves of compound KdV-type equation in the form of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${u_t} + a{u^p}ux + b{u^{2p}}ux + {u_{xxx}} = 0\left( {b \geqslant 0,p{\text{ > }}0} \right)$\end{document}. Our results imply that orbital stability of solitary waves is affected not only by the highest-order nonlinear term \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$b{u^{2p}}ux$\end{document}, but also the nonlinear term \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$a{u^p}ux$\end{document}. For the case of b > 0 and 0 < p ≤ 2, we obtain that the positive solitary wave u1(x−ct) is stable when a > 0, while that unstable when a < 0. The stability for negative solitary wave u2(x − ct) is on the contrary. In particular, we point that the nonlinear term with coefficient a makes contributes to the stability of the solitary waves when p = 2 and a > 0.