The Painlevé integrability and N-solitonic solution in terms of the wronskian determinant for a variable-coefficient variant boussinesq model of nonlinear waves

A variable-coefficient variant Boussinesq (VCVB) model describes the propagation of long waves in shallow water, the nonlinear lattice waves, the ion sound waves in plasmas, and the vibrations in a nonlinear string. With the help of symbolic computation, a VCVB model is investigated for its integrability through the Painlevé analysis. Then, by truncating the Painlevé expansion at the constant level term with two singular manifolds, the dependent variable transformations are obtained through which the VCVB model is bilinearized. Furthermore, the corresponding N-solitonic solutions with graphic analysis are given by the Hirota method and Wronskian technique. Additionally, a bilinear Bäcklund transformation is constructed for the VCVB model, by which a sample one-solitonic solution is presented.