## Abstract

We consider the orbital stability of line solitons of the Kadomtsev-Petviashvili-I equation in R × (R/2πZ). Zakharov [40] and Rousset-Tzvetkov [31] proved the orbital instability of the line solitons of the Kadomtsev-Petviashvili-I equation on R^{2}. The orbital instability of the line solitons on R × (R/2πZ) with the traveling speed c > ^{√}^{4}_{3} was proved by Rousset-Tzvetkov [32] and the orbital stability of the line solitons with the traveling speed 0 < c < ^{√}^{4}_{3} was showed in [34]. In this paper, we prove the orbital stability of the line soliton of the Kadomtsev-Petviashvili-I equation on R × (R/2πZ) with the critical speed c = ^{√}3 4 and the Zaitsev solitons near the line soliton. Since the linearized operator around the line soliton with the traveling speed ^{√}^{4}_{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].

Original language | English |
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Pages (from-to) | 507-526 |

Number of pages | 20 |

Journal | Differential and Integral Equations |

Volume | 33 |

Issue number | 9-10 |

Publication status | Published - Sep 2020 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- Analysis
- Applied Mathematics