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.