Two new integrable fourth-order nonlinear equations: multiple soliton solutions and multiple complex soliton solutions

In this paper, we develop two new fourth-order integrable equations represented by nonlinear PDEs of second-order derivative in time t . The new equations model both right- and left-going waves in a like manner to the Boussinesq equation. We will employ the Painlevé analysis to formally show the complete integrability of each equation. The simplified Hirota’s method is used to derive multiple soliton solutions for this equation. We introduce a complex form of the simplified Hirota’s method to develop multiple complex soliton solutions. More exact traveling wave solutions for each equation will be derived as well.