A bidirectional fifth-order partial differential equation: Lax representation and soliton solutions

In this paper, we study the bidirectional fifth-order partial differential equation uttt−uxxxxt−4(uxut)xx−4(uxuxt)x=0, which was proposed as a new integrable equation by Wazwaz [Phys. Scr. 83, 015012 (2011)]. By means of the prolongation structure method of Wahlquist and Estabrook [J. Math. Phys. 16, 1–7 (1975)], we construct a Lax representation for this equation. This enables us to confirm its integrability and identify it as the potential second-order flow in a Gelfand-Dickey hierarchy. We also use the Darboux transformation and present the multi-soliton solutions in the Wronskian form. Finally, we comment on a more general bidirectional partial differential equation.