@article{Xiong2024, 
author = {Wenzhen Xiong and Yaqing Liu},
title = {Physical significance and periodic solutions of the high-order good Jaulent-Miodek model in fluid dynamics},
year = {2024},
journal = {AIMS Mathematics},
volume = {9},
number = {11},
pages = {31848-31867},
keywords = {the Jaulent-Miodek model, physically significant, Lax pair, Whitham-JM equation, periodic wave},
url = {https://www.sciopen.com/article/10.3934/math.20241530},
doi = {10.3934/math.20241530},
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.}
}