Painlevé Analysis and Auto-Bäcklund Transformation for a General Variable Coefficient Burgers Equation with Linear Damping Term (Journal of Nonlinear Modeling and Analysis, 2024)
This paper investigates a general variable coefficient (gVC) Burgers equation with linear damping term. We derive the Painlevé property of theequation under certain constraint condition of the coefficients. Then we obtain an auto-Bäcklund transformation of this equation in terms of the Painlevé property. Finally, we find a large number of new explicit exact solutions ofthe equation. Especially, infinite explicit exact singular wave solutions are obtained for the first time. It is worth noting that these singular wave solutionswill blow up on some lines or curves in the $(x,t)$ plane. These facts reflectthe complexity of the structure of the solution of the gVC Burgers equationwith linear damping term. It also reflects the complexity of nonlinear wavepropagation in fluid from one aspect.
Keywords: Painlevé property; auto-Bäcklund transformation; a gVC Burgers equation with linear damping term; exact solutions.