Weakly nonlinear analysis for two types of problems in infinitely long vertical channels is performed in the present paper. The first problem corresponds to the case where the base flow is generated by constant temperature difference between the walls of a plane vertical channel. The amplitude evolution equation in this case is the complex Ginzburg-Landau equation. The coefficients of the equation are calculated for the case where the Prandtl number is close to zero. The results of the calculations are compared with available experimental data. Reasonble agreement is found between weakly nonlinear theory and experiments in terms of the structure of the secondary flow: both theory and experiments confirm the presence of secondary flow with stationary convection cells. Weakly nonlinear theory is also used for the solution of the second problem where steady convective flow in the vertical direction in a tall vertical concentric annulus is generated either by the temperature difference between the walls or by internal hear sources distributed within the fluid. Application of weakly nonlinear theory for the case of arbitrary Prandtl number leads to the conclusion that the amplitude evolution equation for the most unstable mode is the complex Ginzburg-Landau equation.