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 Painlevé integrable property of that equation. With the truncated Painlevé expansion and Hirota method, we work out some bilinear forms, bilinear Bäcklund 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.