Dynamics of solutions for nonlinear dispersive equation

非线性色散方程解的动力学

• 批准号: 21540220
• 负责人: MIZUMACHI Tetsu
• 金额: $ 2.83万
• 依托单位: Kyushu University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2009
• 资助国家: 日本
• 起止时间: 2009 至 2012
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-21540220/
• 关键词: 孤立波; 安定性; 非線形分散型方程式; 平面進行波; KP方程式; KP-II方程式; line-soliton; 1次元Benney-Luke; ソリトンの安定性; KP-II; L^2安定性; 一次元非線形シュレディンガー方程式; Fermi-Pasta-Ulamの格子模型; 多ソリトン

I study stability of solitary waves for long wave models using their monotonicity preoperty. I show that modulations of the line soliton for the KP-II equation can be described a system of the Burgers equation and prove the stability of line soliton solution. I also prove stability of multi solitary waves of the Fermi-Pasta-Ulam lattices in the KdV limit and solitary wave solutions of the Benney-Luke equation which is a kind of bidirectional models akin to the Boussinesq systems. In a joint work with Tzvetkov, we show that stability of line soliton solutions for the KP-II equation in L^2(R_x×T_y) is equivalent to the stability of the null solution by using the Miura transformation and obtain L^2(R_x×T_y) stability of line solitons. The idea was tranfered to 1D cubic NLS in a joint work with Pelinovsky.

我利用单调性先验研究了长波模型的孤波稳定性，证明了 KP-II 方程的线孤子调制可以描述为 Burgers 方程组，并证明了线孤子解的稳定性。 Fermi-Pasta-Ulam 晶格在 KdV 极限下的多孤立波稳定性和 Benney-Luke 方程的孤立波解，Benney-Luke 方程是一种类似于在与 Tzvetkov 的合作中，我们通过使用 Miura 变换证明了 L^2(R_x×T_y) 中 KP-II 方程的线孤子解的稳定性相当于零解的稳定性，并得到线孤子的 L^2(R_x×T_y) 稳定性在与 Pelinovsky 的合作中被转移到一维立方 NLS。

项目成果

Stability of co-propagating N-solitons of FPU lattices in the KdV limit

FPU 晶格共传播 N 孤子在 KdV 极限下的稳定性

• 发表时间: 2009
• 作者: Kousuke Kuto;Yoshio Yamada;水町 徹
• 通讯作者: 水町 徹

Asymptotic stability of solitary waves in the Benney-Luke model of water waves

水波 Benney-Luke 模型中孤立波的渐近稳定性

• 发表时间: 2013
• 期刊: Differential and Integral Equations
• 影响因子: 1.4
• 作者: Tetsu Mizumachi;Jos'e Ra'ul Quintero and Robert L. Pego
• 通讯作者: Jos'e Ra'ul Quintero and Robert L. Pego

N-soliton states of the FPU lattices

FPU 晶格的 N 孤子态

• DOI: 10.1137/100792457
• 发表时间: 2011
• 期刊: SIAM Journal on Mathematical Analysis
• 影响因子: 2
• 作者: 高崎金久;T.Takebe;L.P.Teo;T.Mizumachi
• 通讯作者: T.Mizumachi

Stability of the line soliton of the KP-II equation under periodic transverse perturbations

• DOI: 10.1007/s00208-011-0654-3
• 发表时间: 2010-08
• 期刊: Mathematische Annalen
• 影响因子: 1.4
• 作者: Tetsu Mizumachi;N. Tzvetkov

Description of the Inelastic Collision of Two Solitary Waves for the BBM Equation

• DOI: 10.1007/s00205-009-0244-7
• 发表时间: 2008-03
• 期刊: Archive for Rational Mechanics and Analysis
• 影响因子: 2.5
• 作者: Y. Martel;F. Merle;Tetsu Mizumachi