Darboux transformation and multi-soliton solutions of (1+1)-dimensional dispersive long wave equations for water waves

With the aid of symbolic computation, (1+1)-dimensional dispersive long wave equations is proved to possess the Painlevé property by using the WTC method and Kruskal simplification, then based on the constructing Lax pair, the Darboux transformation with multi-parameters for (1+1)-dimensional dispersive long wave equations is presented. As an application, new explicit (2N − 1)-soliton solutions of (1+1)-dimensional dispersive long wave equation are obtained. When N = 2, the properties for three-soliton solutions are graphically studied, which might be helpful to understand the propagation of water waves.