Stability of solitary waves for generalized $abcd$-Boussinesq system: The Hamiltonian case
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| Format: | Preprint |
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2023
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| _version_ | 1866916769447280640 |
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| author | Filho, Roberto de A. Capistrano Quintero, Jose Raul Sun, Shu-Ming |
| author_facet | Filho, Roberto de A. Capistrano Quintero, Jose Raul Sun, Shu-Ming |
| contents | The $abcd$-Boussinesq system is a model of two equations that can describe the propagation of small-amplitude long waves in both directions in the water of finite depth. Considering the Hamiltonian regimes, where the parameters $b$ and $d$ in the system satisfy $b=d>0$, small solutions in the energy space are globally defined. Then, a variational approach is applied to establish the existence and nonlinear stability of the set of solitary-wave solutions for the generalized $abcb$-Boussinesq system. The main point of the analysis is to show that the traveling-wave solutions of the generalized $abcb$-Boussinesq system converge to nontrivial solitary-wave solutions of the generalized Korteweg-de Vries equation. Moreover, if $p$ is the exponent of the nonlinear terms for the generalized $abcb$-Boussinesq system, then the nonlinear stability of the set of solitary-waves is obtained for any $p$ with $ 0 < p < p_0$ where $p_0 $ is strictly larger than $4$, while it has been known that the critical exponent for the stability of solitary waves of the generalized KdV equations is equal to $ 4$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2306_17335 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Stability of solitary waves for generalized $abcd$-Boussinesq system: The Hamiltonian case Filho, Roberto de A. Capistrano Quintero, Jose Raul Sun, Shu-Ming Analysis of PDEs 35B35, 76B25, 35Q35 The $abcd$-Boussinesq system is a model of two equations that can describe the propagation of small-amplitude long waves in both directions in the water of finite depth. Considering the Hamiltonian regimes, where the parameters $b$ and $d$ in the system satisfy $b=d>0$, small solutions in the energy space are globally defined. Then, a variational approach is applied to establish the existence and nonlinear stability of the set of solitary-wave solutions for the generalized $abcb$-Boussinesq system. The main point of the analysis is to show that the traveling-wave solutions of the generalized $abcb$-Boussinesq system converge to nontrivial solitary-wave solutions of the generalized Korteweg-de Vries equation. Moreover, if $p$ is the exponent of the nonlinear terms for the generalized $abcb$-Boussinesq system, then the nonlinear stability of the set of solitary-waves is obtained for any $p$ with $ 0 < p < p_0$ where $p_0 $ is strictly larger than $4$, while it has been known that the critical exponent for the stability of solitary waves of the generalized KdV equations is equal to $ 4$. |
| title | Stability of solitary waves for generalized $abcd$-Boussinesq system: The Hamiltonian case |
| topic | Analysis of PDEs 35B35, 76B25, 35Q35 |
| url | https://arxiv.org/abs/2306.17335 |