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].

中文翻译：

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临界行进速度下Kadomtsev–Petviashvili-I方程的孤子线的稳定性

我们考虑Kadomtsev–Petviashvili-I方程的线孤子在$ \ mathbb R \ times（\ mathbb R / 2 \ pi \ mathbb Z）$中的轨道稳定性。Zakharov [40]和Rousset–Tzvetkov [31]证明了$ \ mathbb R ^ 2 $上Kadomtsev–Petviashvili-I方程的线孤子的轨道不稳定性。行进速度$ c> {\ frac {4} {\ sqrt {3}}} $上$ \ mathbb R \ times（\ mathbb R / 2 \ pi \ mathbb Z）$上的行孤子的轨道不稳定性为由Rousset–Tzvetkov [32]证明，行列孤子在传播速度$ 0 <c <{\ frac {4} {\ sqrt {3}}} $上的轨道稳定性在[34]中得到了证明。在本文中，我们证明了在$ \ mathbb R \ times（\ mathbb R / 2 \ pi \ mathbb Z）$上，Kadomtsev–Petviashvili-I方程的孤子线的轨道稳定性，其临界速度为$ c = {\ frac {4} {\ sqrt {3}}} $和孤子线附近的Zaitsev孤子。由于以行进速度$ {\ frac {4} {\ sqrt {3}}} $绕线孤子的线性化算子是简并的，因此我们无法在[32，33，34]中应用自变量。为了证明稳定性，我们研究了Zaitsev孤子的分支并应用了论点[37]。