Novel complexiton, rational wave, multi-lumps and the kink solitary wave solutions to the new (3+1)-dimensional integrable fourth-order equation for shallow water waves

Abstract In this exploration, we aim to seek a number of new exact solutions to the new (3+1)-dimensional integrable fourth-order nonlinear equation, which is widely used to describe the shallow water waves. Employing the Cole-Hopf transformation, we develop its bilinear form. Then, taking advantage of the ansatz function method, a new functional form is utilized to probe the singular complexiton solutions. Based on which, the non-singular complexiton solutions are derived by imposing the constraint conditions. In addition, we find the rational wave solutions and multi-lumps solutions wielding the rational function method and new homoclinic method respectively. At the end, we investigate the kink solitary wave solutions using the variational approach that is based on the variational principle and Ritz method. Meanwhile, the Hamiltonian of the system is also elaborated. Correspondingly, the graphic descriptions of the extracted results are presented to unfold their dynamic behaviors through Maple. As we all know, the findings of this paper are firstly reported and can enlarge the exact solutions of the considered PDE.