Integrability for the generalised variable-coefficient fifth-order Korteweg–de Vries equation with Bell polynomials

In the present paper, a direct and unifying Bell polynomial approach is employed to find the bilinear representations, bilinear Bäcklund transformation and Lax pair of a generalised variable-coefficient fifth-order Korteweg–de Vries equation. Note that the integrable constraint conditions on the variable coefficients can be naturally found in the procedure of applying the Bell polynomials. In addition, with the help of the Hirota bilinear method, the N -soliton solutions of this equation are also obtained.