Methods of weakly nonlinear theory are used in the present paper in order to analyze the behavior of the most unstable mode above the threshold. Equations of motion are shallow water equations for two-phase slightly curved mixing layers under three simplifying assumptions: (a) rigid-lid assumption, (b) homogeneous distribution of particles in the carrier fluid and (c) absence of dynamical interaction between particles and the carrier fluid. Two approaches are investigated: parallel base flow assumption and slow variation of the base flow in the longitudinal direction. The complex Ginzburg-Landau equation for the amplitude of the most unstable mode is derived in the first case. First-order partial differential equation for the amplitude is derived in the second case. In both cases the coefficients of the equations can be calculated in terms of integrals containing the linearized characteristics of the problem.