Integrable properties of a variable-coefficient Korteweg–de Vries model from Bose–Einstein condensates and fluid dynamics

The phenomena of the trapped Bose–Einstein condensates related to matter waves and nonlinear atom optics can be governed by a variable-coefficient Korteweg–de Vries (vc-KdV) model with additional terms contributed from the inhomogeneity in the axial direction and the strong transverse confinement of the condensate, and such a model can also be used to describe the water waves propagating in a channel with an uneven bottom and/or deformed walls. In this paper, with the help of symbolic computation, the bilinear form for the vc-KdV model is obtained and some exact solitonic solutions including the N-solitonic solution in explicit form are derived through the extended Hirota method. We also derive the auto-Bäcklund transformation, nonlinear superposition formula, Lax pairs and conservation laws of this model. Finally, the integrability of the variable-coefficient model and the characteristic of the nonlinear superposition formula are discussed.