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In this work, we explore the Riemann problem of some integrable equations given initial data in the form of a step discontinuity by employing the Whitham modulation theory. The periodic wave solutions of the equation are characterized by elliptic functions along with the Whitham modulation equations. Moreover, we find that the $\pm$ signs for the velocities of the periodic wave solutions remain unchanged during propagation.hus, when analyzing the propagation behavior of solutions, it is necessary to separately consider the clockwise (negative velocity) and counterclockwise (positive velocity) cases. In this regard, we present the classification of the solutions to the Riemann problem of the equation in both clockwise and counterclockwise cases for the first time. 更多阅读