Dynamics of the (2+1)$$ \left(2+1\right) $$‐dimensional coupled cubic–quintic complex Ginzburg–Landau model for convecting binary fluid: Analytical investigations and multiple‐soliton solutions

Abstract

The ‐dimensional coupled cubic–quintic complex Ginzburg–Landau equations ( ‐DCC‐QCGLEs) can simulate a variety of binary fluid thermal convection characteristics, containing complex parameters. The analysis of pattern formation in chaotic and nonlinear dynamical systems can benefit greatly from the use of this thermal convection model. The primary goal of this study is to find analytical solutions for some recent advances that have been made for Rayleigh–Bénard convection by applying the ‐DCC‐QCGLEs model for slowly varying spatio‐temporal amplitudes of the wave motion. In addition, novel traveling solitary wave solutions for the model equation are derived using a very useful method to investigate how complex physical coefficients affect the profiles of propagating waves. Furthermore, we introduce the WTC‐Kruskal algorithm of the Painlevé methodology to examine the integrability of the ‐DCC‐QCGLEs and the truncated Painlevé expansion is used to extract the Bäcklund transform, from which new solitary solutions can be acquired. The results also demonstrated a good agreement with previous works and were more significant and accurate in two and three dimensions of the proposed model. Finally, the computational results indicate that the effects of the physical parameters of the considered equations can be demonstrated by utilizing 2D and 3D graphics for different values of these parameters.