Painlevé Analysis, Bilinear Forms, Bäcklund Transformations and Solitons for a Variable-Coefficient Extended Korteweg-de Vries Equation with an External-Force Term in Fluid Mechanics and Plasma Dynamics

In this paper, we investigate a variable-coefficient extended Korteweg-de Vries equation with an external-force term in fluid mechanics and plasma dynamics. Under certain variable-coefficient constraints, we get the Painlev & eacute; integrable property of that equation. With the truncated Painlev & eacute; expansion and Hirota method, we work out some bilinear forms, bilinear B & auml;cklund transformations under certain variable-coefficient constraints. With the bilinear forms, multi-soliton solutions are constructed. Based on those solutions, multi-complex-soliton solutions are derived through the complex forms of the Hirota method. Influences of the variable coefficients on the multi-soliton solutions are discussed graphically. We find that (i) different types of the one-soliton profiles and soliton interactions can be seen with the changes of variable coefficients; (ii) the amplitudes of those solitons are influenced under the dissipative and cubic-nonlinear coefficients; (iii) the characteristic lines and velocities of those solitons are influenced under the dissipative, dispersive coefficients and external-force term; (iv) the backgrounds of those solitons are influenced under the external-force term. Additionally, the influences of the variable coefficients on the complex solitons are similar to the influences on the real solitons.