The transition process of dune formation is known to show a hysteresis that the Froude number at which dunes disappear with increasing discharge is larger than the Froude number at which dunes reappear with decreasing discharge. In this study, a weakly nonlinear stability analysis is performed in order to provide a theoretical explanation for the hysteresis observed in the dune transition. The growth rate expansion method with the aid of the multiple scale expansion technique is used to derive a nonlinear amplitude equation, the Landau equation, from two-dimensional Reynolds equations and the Exner equation for sediment conservation with five boundary conditions at the water and bed surfaces. The results show that the first Landau constant is positive in the range of parameters employed in this analysis; thus, the transition from dune-covered to flat beds corresponds to the subcritical bifurcation. This implies that there is a great possibility that the hysteresis is caused by the subcritical bifurcation.
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