Painlevé analysis, auto-Bäcklund transformations and exact solutions for a simplified model for reacting mixtures

Recently, a simplified model for reacting mixtures was investigated to admit the potential symmetries only when the structure function f(u) in the model was of the form u/2βγ+k. In this paper, we investigate some other properties of the nonlinear model with the condition f(u)=u/2βγ+k. Firstly, the Painlevé analysis is performed such that it is shown that this equation passes the Painlevé test. And then two new types of auto-Bäcklund transformations (ABTs) are found by using the truncated Painlevé expansion analysis and some ansatz. The ABTs reduce the nonlinear model to the systems of linear partial differential equations with respect to the new introduced variables. Finally, based on the obtained auto-Bäcklund transformations, we explore some explicit exact solutions including soliton solutions, singular soliton solutions soliton-like solutions, rational solutions and other types of solutions, which may be useful to explain the corresponding physical phenomena.