Painlevé analysis, Painlevé–Bäcklund, multiple regular and singular kink solutions of dynamical thermopherotic equation drafting wrinkle propagation

The thermophoretic motion (TM) system with a variable heat transmission factor, based on the Korteweg-de Vries (KdV) equation, is used to model soliton-like thermophoresis of creases in graphene sheets. Painlevé test is employed to discover that the equation is Painlevé integrable. Then an auto-Bäcklund transformation using the truncated Painlevé expansion is obtained. Concerning the additional variables, the auto-Bäcklund transformations convert the nonlinear model to a set of linear partial differential equations. Finally, various explicit precise solutions based on the acquired auto-Bäcklund transformations are investigated and the researched solutions are illustrated in 3D, 2D and contour plots. Furthermore, the Cole-Hopf transformation is used in conjunction with Hirota’s bilinear technique to get multiple regular and singular kink solutions.