Stability of the line soliton of the Kadomtsev--Petviashvili-I equation with the critical traveling speed

Abstract

We consider the orbital stability of solitons of the Kadomtsev--Petviashvili-I equation in $\mathbb{R} \times (\mathbb{R}/2\pi\mathbb{Z})$ which is one of a high dimensional generalization of the Korteweg--de Vries equation. Benjamin showed that the Korteweg--de Vries equation possesses the stable one soliton. We regard the one soliton of the Korteweg--de Vries equation as a line soliton of the Kadomtsev--Petviashvili-I equation. Zakharov and Rousset--Tzvetkov proved the orbital instability of the line solitons of the Kadomtsev--Petviashvili-I equation on $\mathbb{R}^2$. In the case of the Kadomtsev--Petviashvili-I equation on $\mathbb{R} \times (\mathbb{R}/2\pi\mathbb{Z})$, the orbital instability of the line solitons with the traveling speed $c>4/\sqrt{3}$ and the orbital stability of the line solitons with the traveling speed $0<c<4/\sqrt{3}$ was proved by Rousset--Tzvetkov. In this paper, we prove the orbital stability of the line soliton of the Kadomtsev--Petviashvili-I equation on $\mathbb{R} \times (\mathbb{R}/2\pi\mathbb{Z})$ with the critical speed $c=4/\sqrt{3}$ and the Zaitsev solitons near the line soliton. Since the linearized operator around the line soliton with the traveling speed $4/\sqrt{3}$ is degenerate, we can not apply the argument by Rousset--Tzvetkov. To prove the stability of the line soliton, we investigate the branch of the Zaitsev solitons.