Painlevé integrable property, Bäcklund transformations, Lax pair, and soliton solutions of a (3 + 1)-dimensional variable-coefficient Hirota bilinear system in a fluid

In this paper, we focus our attention on a (3 + 1)-dimensional variable-coefficient Hirota bilinear system in a fluid with symbolic computation. The Painlevé integrable property is derived. Via the Ablowitz–Kaup–Newell–Segur procedure, we obtain a Lax pair under the coefficient constraints. Based on the Hirota method, we obtain a bilinear form and a bilinear Bäcklund transformation under the coefficient constraints. We derive the auto-Bäcklund transformations based on the truncated Painlevé expansions. According to the bilinear form, we give the two-soliton solutions under the coefficient constraints. We also discuss the relation between the variable coefficients and soliton solutions, i.e., how the two solitons present different types with the different forms of the variable coefficients.