Physical significance and periodic solutions of the high-order good Jaulent-Miodek model in fluid dynamics

Wenzhen Xiong; Yaqing Liu; Wenzhen Xiong; Yaqing Liu

doi:10.3934/math.20241530

AIMS Mathematics

2024, Volume 9, Issue 11: 31848-31867. doi: 10.3934/math.20241530

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Research article

Physical significance and periodic solutions of the high-order good Jaulent-Miodek model in fluid dynamics

Wenzhen Xiong ^{1
,
,},
Yaqing Liu ²

1.
School of Mathematics and Information Engineering, Xinyang Vocational and Technical College, Xinyang 464000, China
2.
School of Applied Science, Beijing Information Science and Technology University, Beijing 100192, China

Received: 03 September 2024 Revised: 27 October 2024 Accepted: 04 November 2024 Published: 08 November 2024
MSC : 35Q31, 35Q51

Using Whitham modulation theory, this paper examined periodic solutions and the problem of discontinuous initial values for the higher-order good Jaulent-Miodek (JM) equation. The physical significance of the JM equations was discussed by considering the reduction of Euler's equation. Next, the zero- and one-phase periodic solutions of the JM equation, along with the associated Whitham equations, were derived. The analysis included the degeneration of the one-phase periodic solution and the genus-one Whitham equation by examining the limits of the modulus $ m $ of the Jacobi elliptic functions. Additionally, analytical and graphical representations of rarefaction wave solutions and periodic wave patterns were provided, and a solution for discontinuous initial values in the JM equation was presented. The results of this study offer a theoretical foundation for analyzing discontinuous initial values in nonlinear dispersion equations.
- the Jaulent-Miodek model,
- physically significant,
- Lax pair,
- Whitham-JM equation,
- periodic wave
Citation: Wenzhen Xiong, Yaqing Liu. Physical significance and periodic solutions of the high-order good Jaulent-Miodek model in fluid dynamics[J]. AIMS Mathematics, 2024, 9(11): 31848-31867. doi: 10.3934/math.20241530

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Abstract

Using Whitham modulation theory, this paper examined periodic solutions and the problem of discontinuous initial values for the higher-order good Jaulent-Miodek (JM) equation. The physical significance of the JM equations was discussed by considering the reduction of Euler's equation. Next, the zero- and one-phase periodic solutions of the JM equation, along with the associated Whitham equations, were derived. The analysis included the degeneration of the one-phase periodic solution and the genus-one Whitham equation by examining the limits of the modulus $ m $ of the Jacobi elliptic functions. Additionally, analytical and graphical representations of rarefaction wave solutions and periodic wave patterns were provided, and a solution for discontinuous initial values in the JM equation was presented. The results of this study offer a theoretical foundation for analyzing discontinuous initial values in nonlinear dispersion equations.

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