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We consider the KP-I and gKP-I equations in $\mathbb{R}\times(\mathbb{R}/2\pi \mathbb{Z})$. We prove that the KdV soliton with subcriticalspeed $0<c<c^*$ is orbitally stable under the global KP-I flow constructed byIonescu and Kenig \cite{IK}. For supercritical speeds $c>c^*$, in the spirit ofthe work by Duyckaerts and Merle \cite{DM}, we sharpen our previous instabilityresult and construct a global solution which is different from the solitarywave and its translates and which converges to the solitary wave as time goesto infinity. This last result also holds for the gKP-I equation.

Stability and instability of the KdV solitary wave under the KP-I flow