Integrability of the modified generalised Vakhnenko equation

Integrability of the modified generalised Vakhnenko equation is investigated systematically. Based on binary Bell polynomials, its bilinear representation, N soliton solutions, bilinear Bäcklund transformation, and Lax pair are succinctly constructed. Moreover, the conservation laws of the modified generalised Vakhnenko equation are discussed by using corresponding Lax pair. Furthermore, the quasiperiodic solution of the modified generalised Vakhnenko equation is presented by applying Hirota direct method and Riemann theta function. The asymptotic behavior of the one periodic wave is analyzed in details. It is shown that the one periodic wave solution tends to the one soliton solution under a small amplitude limit λ → 0. Finally, the new N soliton solutions of the standard Vakhnenko equation are presented. It would be specially mentioned that all the results of modified generalised Vakhnenko equation can be reduced to the generalised Vakhnenko equation and standard Vakhnenko equation under the special case of α = 1 and α = 1, β = 0, respectively.