Lax pair and Darboux transformation for a variable-coefficient fifth-order Korteweg-de Vries equation with symbolic computation

In this paper, we put our focus on a variable-coefficient fifth-order Korteweg-de Vries (fKdV) equation which possesses a great of excellent properties and is of current importance in physical and engineering fields. Certain constraints are worked out, which make sure the integrability of such an equation. Under those constraints, some integrable properties are derived such as the Lax pair and Darboux transformation. Via the Darboux transformation, which is an exercisable way to generate solutions in a recursive manner, the one- and two-solitonic solutions are presented and the relevant physical applications of these solitonic structures in some fields are also pointed out.