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December 2022
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Research Article| December 01 2022
Deng-Shan Wang
;
Deng-Shan Wang a)
(Conceptualization, Formal analysis, Investigation, Methodology, Software, Writing – original draft, Writing – review & editing)
Laboratory of Mathematics and Complex Systems (Ministry of Education), School of Mathematical Sciences, Beijing Normal University
, Beijing 100875,
China
a)Author to whom correspondence should be addressed: dswang@bnu.edu.cn
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Xiaodong Zhu
Xiaodong Zhu
(Formal analysis, Investigation)
Laboratory of Mathematics and Complex Systems (Ministry of Education), School of Mathematical Sciences, Beijing Normal University
, Beijing 100875,
China
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Author & Article Information
a)Author to whom correspondence should be addressed: dswang@bnu.edu.cn
J. Math. Phys. 63, 123501 (2022)
Article history
Received:
August 04 2022
Accepted:
November 08 2022
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Citation
Deng-Shan Wang, Xiaodong Zhu; Long-time asymptotics of the good Boussinesq equation with qxx-term and its modified version. J. Math. Phys. 1 December 2022; 63 (12): 123501. https://doi.org/10.1063/5.0118374
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Two modified Boussinesq equations along with their Lax pairs are proposed by introducing the Miura transformations. The modified good Boussinesq equation with initial condition is investigated by the Riemann–Hilbert method. Starting with the three-order Lax pair of this equation, the inverse scattering transform is formulated and the Riemann–Hilbert problem is established, and the properties of the reflection coefficients are presented. Then, the formulas of long-time asymptotics to the good Boussinesq equation and its modified version are given based on the Deift–Zhou approach of nonlinear steepest descent analysis. It is demonstrated that the results from the long-time asymptotic analysis are in excellent agreement with the numerical solutions. This is the first result on the long-time asymptotic behaviors of the good Boussinesq equation with qxx-term and its modified version.
Topics
Soliton solutions, Asymptotic analysis, Inverse scattering, Jost functions
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© 2022 Author(s). Published under an exclusive license by AIP Publishing.
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