Miura transformations and large-time behaviors of the Hirota-Satsuma equation
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929500544040960 |
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| author | Wang, Deng-Shan Zhu, Cheng Zhu, Xiaodong |
| author_facet | Wang, Deng-Shan Zhu, Cheng Zhu, Xiaodong |
| contents | The good Boussinesq equation has several modified versions such as the modified Boussinesq equation, Mikhailov-Lenells equation and Hirota-Satsuma equation. This work builds the full relations among these equations by Miura transformation and invertible linear transformations and draws a pyramid diagram to demonstrate such relations. The direct and inverse spectral analysis shows that the solution of Riemann-Hilbert problem for Hirota-Satsuma equation has simple pole at origin, the solution of Riemann-Hilbert problem for the good Boussinesq equation has double pole at origin, while the solution of Riemann-Hilbert problem for the modified Boussinesq equation and Mikhailov-Lenells equation doesn't have singularity at origin. Further, the large-time asymptotic behaviors of the Hirota-Satsuma equation with Schwartz class initial value is studied by Deift-Zhou nonlinear steepest descent analysis. In such initial condition, the asymptotic expressions of the Hirota-Satsuma equation and good Boussinesq equation away from the origin are derived and it is shown that the leading term of asymptotic formulas matches well the direct numerical simulations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_01215 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Miura transformations and large-time behaviors of the Hirota-Satsuma equation Wang, Deng-Shan Zhu, Cheng Zhu, Xiaodong Exactly Solvable and Integrable Systems The good Boussinesq equation has several modified versions such as the modified Boussinesq equation, Mikhailov-Lenells equation and Hirota-Satsuma equation. This work builds the full relations among these equations by Miura transformation and invertible linear transformations and draws a pyramid diagram to demonstrate such relations. The direct and inverse spectral analysis shows that the solution of Riemann-Hilbert problem for Hirota-Satsuma equation has simple pole at origin, the solution of Riemann-Hilbert problem for the good Boussinesq equation has double pole at origin, while the solution of Riemann-Hilbert problem for the modified Boussinesq equation and Mikhailov-Lenells equation doesn't have singularity at origin. Further, the large-time asymptotic behaviors of the Hirota-Satsuma equation with Schwartz class initial value is studied by Deift-Zhou nonlinear steepest descent analysis. In such initial condition, the asymptotic expressions of the Hirota-Satsuma equation and good Boussinesq equation away from the origin are derived and it is shown that the leading term of asymptotic formulas matches well the direct numerical simulations. |
| title | Miura transformations and large-time behaviors of the Hirota-Satsuma equation |
| topic | Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/2404.01215 |