Orbital stability and strong instability of solitary waves for the Kadomtsev–Petviashvili equation with combined power nonlinearities

Abstract

In this paper, we study the orbital stability and strong instability of solitary waves for the two-dimensional Kadomtsev–Petviashvili equation with combined power nonlinearities ${({\varphi }_t+{\varphi }_{xxx}+{(\mu {\left|\varphi \right|}^{p_1-1}\varphi +{\left|\varphi \right|}^{p_2-1}\varphi )}_x)}_x={\varphi }_{yy},$where $\mu >0$. When $1<p_1<p_2<\frac{7}{3}$, we show that the set of solitary wave solutions is orbitally stable by using variational methods. When $1<p_1<\frac{7}{3}$ and $3\le p_2<5$, we establish that all energy minimizers correspond to local minima of the associated energy functional and then the set of energy minimizers is orbitally stable. Furthermore, we show that the solitary waves are strongly unstable in the case of $3\le p_1<p_2<5$ by blow-up. Our main results improve and complement the existing results in the literature recently.