Large-scale longitudinal distortions of Marangoni wave patterns in the non-isothermal liquid layer covered by surfactant

In the present paper, we consider single traveling waves (STW) generated by the oscillatory instability of Marangoni convection in the thin non-isothermal liquid layer with deformable free surface. The layer is covered by insoluble surfactant that plays an active role in the pattern selection together with inhomogeneity of temperature along the interface and surface deformability. Using the weakly nonlinear analysis we derived the modified complex Ginzburg-Landau equation describing the large-scale distortions of STWs near the bifurcation point. Linear stability analysis reveals existence of two modulational modes: one is for the amplitude and another one for the phase (Benjamin-Feir). Numerically, we found that STWs are stable with respect to longitudinal modulations in the case without surfactant. In the presence of the insoluble surfactant both modulational modes are found. The stability maps for different values of the surfactant concentration are plotted.