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

Abstract

We consider the orbital stability of line solitons of the Kadomtsev–Petviashvili-I equation in $\mathbb R \times (\mathbb R/2\pi\mathbb Z)$. Zakharov [40] and Rousset–Tzvetkov [31] proved the orbital instability of the line solitons of the Kadomtsev–Petviashvili-I equation on $\mathbb R^2$. The orbital instability of the line solitons on $\mathbb R \times (\mathbb R/2\pi\mathbb Z)$ with the traveling speed $c > {\frac {4}{\sqrt{3}}} $ was proved by Rousset–Tzvetkov [32] and the orbital stability of the line solitons with the traveling speed $0 < c < {\frac {4}{\sqrt{3}}} $ was showed in [34]. 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= {\frac {4}{\sqrt{3}}} $ and the Zaitsev solitons near the line soliton. Since the linearized operator around the line soliton with the traveling speed $ {\frac {4}{\sqrt{3}}} $ is degenerate, we cannot apply the argument in [32, 33, 34]. To prove the stability, we investigate the branch of the Zaitsev solitons and apply the argument [37].