Multi-soliton solutions of the forced variable-coefficient extended Korteweg–de Vries equation arisen in fluid dynamics of internal solitary waves

Under investigation in this paper, with symbolic computation, is a forced variable-coefficient extended Korteweg–de Vries equation, which can describe the weakly-nonlinear long internal solitary waves (ISWs) in the fluid with the continuous stratification on density. By virtue of the Hirota bilinear method, multi-soliton solutions for such an equation with the external force term have been derived. Furthermore, effects are discussed with the aid of the characteristic line: (I) Inhomogeneities of media and nonuniformities of boundaries, depicted by the variable coefficients, play a role in the soliton behavior; (II) Solitons change their initial propagation direction on the compact shock or anti-shock wave background in the presence of the external time-dependent force, and the results present an extended view compared with that for the linear theory; (III) Combined effects of the inhomogeneities and external force are regarded as the nonlinear composition of the independent influence induced by the two factors. Those results could be expected to be helpful for the investigation on the dynamics of the ISWs in an ocean or atmosphere stratified fluid.