Longitudinal Modulation of Marangoni Wave Patterns in Thin Film Heated From Below: Instabilities and Control

Non-linear Marangoni waves, which are generated by the long-wave oscillatory instability of the conductive state in a thin liquid film heated from below in the case of a deformable free surface and a substrate of very low conductivity, are considered. Previously, the investigation of traveling Marangoni waves was restricted to the analysis of the bifurcation and stability with respect to disturbances with strongly different wave vectors. In the present article, for the first time, the modulational instability of traveling waves is investigated. We derive the amplitude equation for the modulated traveling wave, which describes non-linear interaction of the main convective pattern with the perturbations with slightly different wavenumbers. The amplitude equation differs from the conventional complex Ginzburg–Landau equation as it contains an additional term of the local liquid level rise. Linear stability analysis reveals two modulational instability modes: the amplitude modulational and the phase modulational (Benjamin–Feir) ones. It is shown that traveling rolls are stable against the longitudinal modulation for the uncontrolled convection. We also investigate the influence of the non-linear feedback control, which was applied previously to eliminate subcritical excitation of traveling rolls. Computations reveal both the modulational modes under the non-linear feedback control. The obtained results show that the modulational instabilities significantly influence the region of parameters where the non-linear feedback control is efficient for stabilization of waves.

Families of Solutions of Multitemporal Nonlinear Schrödinger PDE

The multitemporal nonlinear Schrödinger PDE (with oblique derivative) was stated for the first time in our research group as a universal amplitude equation which can be derived via a multiple scaling analysis in order to describe slow modulations of the envelope of a spatially and temporarily oscillating wave packet in space and multitime (an equation which governs the dynamics of solitons through meta-materials). Now we exploit some hypotheses in order to find important explicit families of exact solutions in all dimensions for the multitime nonlinear Schrödinger PDE with a multitemporal directional derivative term. Using quite effective methods, we discovered families of ODEs and PDEs whose solutions generate solutions of multitime nonlinear Schrödinger PDE. Each new construction involves a relatively small amount of intermediate calculations.

Amplitude equation formalism for reaction—subdiffusion systems

The paper presents derivation of the amplitude equation for the Hopf bifurcation in the two-component system with nonlinear chemical kinetics and subdiffusion. Anomalous diffusion transport is described via Caputo fractional derivatives. The obtained amplitude equation is much more complex compared to the case of normal diffusion because solutions of fractional order linear differential equations have inconvenient behavior.

A Nonlinear Multiscale Theory of Atmospheric Blocking: Eastward and Upward Propagation and Energy Dispersion of Tropospheric Blocking Wave Packets

In this paper, a nonlinear multiscale interaction model is used to examine how the planetary waves associated with eddy-driven blocking wave packets propagate through the troposphere in vertically varying weak baroclinic basic westerly winds (BWWs). Using this model, a new one-dimensional finite-amplitude local wave activity flux (WAF) is formulated, which consists of linear WAF related to linear group velocity and local eddy-induced WAF related to the modulus amplitude of blocking envelope amplitude and its zonal nonuniform phase. It is found that the local eddy-induced WAF reduces the divergence (convergence) of linear WAF in the blocking upstream (downstream) side to favor blocking during the blocking growth phase. But during the blocking decay phase, enhanced WAF convergence occurs in the blocking downstream region and in the upper troposphere when BWW is stronger in the upper troposphere than in the lower troposphere, which leads to enhanced upward-propagating tropospheric wave activity, though the linear WAF plays a major role. In contrast, the downward propagation of planetary waves may be seen in the troposphere for vertically decreased BWWs. These are not seen for a zonally uniform eddy forcing. A perturbed inverse scattering transform method is used to solve the blocking envelope amplitude equation. It is found that the finite-amplitude WAF represents a modified group velocity related to the variations of blocking soliton amplitude and zonal wavenumber caused by local eddy forcing. Using this amplitude equation solution, it is revealed that, under local eddy forcing, the blocking wave packet tends to be nearly nondispersive during its growth phase but strongly dispersive during the decay phase for vertically increased BWWs, leading to strong eastward and upward propagation of planetary waves in the downstream troposphere.

Study on Turing Patterns of Gray–Scott Model via Amplitude Equation

In this paper, the amplitude equations of a Gray–Scott model without (or with) the feedback time delay are derived based on weakly nonlinear method, by which the selection of Turing patterns for this model can be theoretically determined. As a result, the effects of the diffusion coefficient ratio and the time delay factor on the Turing pattern can be investigated as the main purpose of this paper. If one of the diffusion coefficients is chosen as the bifurcation control parameter in the procedure of the amplitude equation at first, it is proved that the first-order bifurcation of the Turing patterns is only determined by the diffusion coefficient ratio and independent of the concrete value of each diffusion coefficient once the parameters of the reaction terms are fixed as the appropriate constants in the regions of Turing patterns. Furthermore, the feedback time delay factor has no effect on the first-order bifurcation of the Turing patterns, but affects the morphological characteristics of the Turing patterns, especially in the case of large ratio of the diffusion coefficients. With time increasing, the feedback time delay factor can postpone the formation of the Turing patterns and cause the oscillations of Turing patterns at each spatial position. By implementing the numerical calculations for this model, the various Turing patterns with different values of the diffusion coefficient ratios are presented, which really verify the dependence of the diffusion coefficient ratio and independence of the feedback time delay on the first-order bifurcation of the Turing patterns.