Numerical investigation on square Turing patterns in medium with two coupled layers

Abstract

In this paper, the simple and superlattice square patterns in two-dimensional space are investigated numerically by the two-layer coupled Lengyel-Epstein model. When the wave number ratio of Turing modes is greater than one, our results show that the spatial resonance form of the fundamental mode is changed with the increase of coupling strength, and simple hexagon pattern evolves spontaneously into a new pattern with a complicated structure. In addition to the reported superlattice hexagonal pattern, simple square pattern and superlattice square pattern are obtained, such as the complicated big-small spot, spot-line, ring and white-eye square pattern. The characteristics of simple and complicated superlattice square pattern are investigated by the intermediate process of evolution. When the coupling parameters <inline-formula><tex-math id="M7">\begin{document}$\alpha $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182167_M7.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182167_M7.png"/></alternatives></inline-formula> and <inline-formula><tex-math id="M8">\begin{document}$\beta $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182167_M8.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182167_M8.png"/></alternatives></inline-formula> increase synchronously within a certain range, the type I square patterns of the same wavelength are obtained in the two subsystems. When the coupling parameters <inline-formula><tex-math id="M9">\begin{document}$\alpha $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182167_M9.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182167_M9.png"/></alternatives></inline-formula> and <inline-formula><tex-math id="M10">\begin{document}$\beta $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182167_M10.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182167_M10.png"/></alternatives></inline-formula> increase asynchronously, the type I square pattern can evolve into the type II square pattern on the same spatial scale through phase transition. Then, the new subharmonic modes are generated, and the complicated superlattice square patterns are obtained due to the resonance between the two Turing modes in a short wavelength mode subsystem. The influence of coupling between two subsystems on the square pattern is investigated. When the type I square pattern of wavelength <inline-formula><tex-math id="M11">\begin{document}$\lambda $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182167_M11.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182167_M11.png"/></alternatives></inline-formula> emerges, the square pattern will quickly lose its stability in the short wavelength mode subsystem, since the coupling coefficient is equal to zero. Finally a new square pattern of wavelength <inline-formula><tex-math id="M12">\begin{document}$\lambda $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182167_M12.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20182167_M12.png"/></alternatives></inline-formula>/<i>N</i> is formed. The type I square patterns of two subsystems successively evolve into the type II square patterns through the phase transition. The spots move relatively with the extension of simulation time, and a new mode is generated and forms three-wave resonance in two subsystems, and then the hexagonal pattern dominates the system. Our results also show that the type II square pattern spontaneously transforms into a hexagonal pattern.