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The main work of this paper is to construct the Wronskian solution and investigate the integrability characteristics of the (2 + 1)-dimensional Konopelchenko–Dubrovsky equation. Firstly, the Wronskian technique is used to acquire a sufficient condition of the Wronskian solution. According to the Wronskian form, the soliton solution is obtained by selecting the elements in the determinant that satisfy the linear partial differential systems. Secondly, the bilinear Bäcklund transformation and Bell-polynomial-typed Bäcklund transformation are derived directly via the Hirota bilinear method and the Bell polynomial theory, respectively. Finally, Painlevé analysis proves that this equation possesses the Painlevé property, and a Painlevé-typed Bäcklund transformation is constructed to solve a family of exact solutions by selecting appropriate seed solution. It shows that the Wronskian technique, Bäcklund transformation, Bell polynomial and Painlevé analysis are applicable to obtain the exact solutions of the nonlinear evolution equations, e.g., soliton solution, single-wave solution and two-wave solution. [ABSTRACT FROM AUTHOR]

Variable-coefficient equations can be used to describe certain phenomena when inhomogeneous media and nonuniform boundaries are taken into consideration. Describing the fluid dynamics of shallow-water wave in an open ocean, a (2 + 1)-dimensional generalized variable-coefficient Hirota–Satsuma–Ito equation is investigated in this paper. The integrability is first examined by the Painlevé analysis method. Secondly, the one-soliton and two-soliton solutions and lump solutions of the (2 + 1)-dimensional generalized variable-coefficient Hirota–Satsuma–Ito equations are derived by virtue of the Hirota bilinear method. In the exact solutions, parameter values and variable-coefficient functions are chosen and analyzed for different effects on the shallow-water waves. [ABSTRACT FROM AUTHOR]