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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.

Applicable in fluid dynamics and plasmas, a generalized variable-coefficient Korteweg–de Vries (vcKdV) equation is investigated analytically employing the Hirota bilinear method in this paper. The bilinear form for such a model is derived through a dependent variable transformation. Based on the bilinear form, the integrable properties such as the N-solitonic solution, the Bäcklund transformation and the Lax pair for the vcKdV equation are obtained. Additionally, it is shown that the bilinear Bäcklund transformation can turn into the one denoted in the original variables.