On the Integrability of a Generalized Variable-Coefficient Forced Korteweg-de Vries Equation in Fluids

With the inhomogeneities of media taken into account, under investigation is hereby a generalized variable-coefficient forced Korteweg-de Vries (vc-fKdV) equation, which describes shallow-water waves, internal gravity waves, etc. Under an integrable constraint condition on the variable coefficients, in this paper, the complete integrability of the generalized vc-fKdV equation is proposed. By virtue of a generalization of Bells polynomials, we systematically present its bilinear representations, Backlund transformations, Lax pairs and Darboux covariant Lax pairs, which can be reduced to the ones of some integrable models, such as vcKdV model, cylindrical KdV equation, and an analytical model of tsunami generation. It is very interesting that its bilinear formulism is free for the integrable constraint condition. Besides, researching the Lax equations yield its infinitely conservation laws, all conserved densities and fluxes of them are obtained by explicit recursion formulas. Furthermore, by considering its bilinear formulism with an extra auxiliary variable, we present the soliton solutions and Riemann theta function periodic wave solutions of the equation. According to the constraint among the nonlinear, dispersive, and line-damping coefficients, we further discuss the solitonic structures and interaction properties by some graphic analysis. Finally, the relationships between the periodic wave solutions and soliton solutions are presented in detail by a limiting procedure.