Integrable aspects and soliton interaction for a generalized inhomogeneous Gardner model with external force in plasmas and fluids

A generalized inhomogeneous Gardner model with an external force term is investigated which can govern the soliton propagation and interaction in the vicinity of the negative ion critical density for certain plasmas or of equal layer depths for stratified fluids. Integrable aspects including the Lax pair and the Γ-Riccati-type Bäcklund transformation (Γ-R BT) are presented under the Painlevé conditions. By virtue of the Γ-R BT, analytic one- or two-soliton-like solutions with the inhomogeneous coefficients, external force term, eigenvalue in the Lax pair, and another parameter are obtained. Analytic analysis and graphic illustration imply that (1) the amplitude of a soliton is influenced by the quadratic and cubic nonlinear coefficients, the linear-damping coefficient, and the aforementioned eigenvalue; (2) the solitonic velocity is "controlled" by the inhomogeneous coefficients, the external force term, and the aforementioned eigenvalue and parameter; (3) the solitonic background is affected by the linear-damping coefficient, the external force term and the aforementioned parameter; and (4) the possibility of solitonic infection is dominated by the difference between eigenvalues.