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In this paper, exp-expansion method, modified extended tanh-function method and Kudryashov method are used to investigate the exact solutions of the coupled Benjamin–Bona–Mahony–KdV (BBM-KdV) system. By the travelling wave transformation, the coupled BBM-KdV system is reduced to ordinary differential equations. Solving the nonlinear ordinary differential equations, kink, periodic and solitary solutions of the system are obtained. To further understand the natures of the solutions more intuitively, the dynamical behaviours of the solutions are given. The physical structures of the solutions are analysed in combination with 2D and 3D plots. These results enrich the available types of travelling wave solutions for the BBM-KdV model and are also useful for wave studies in complex engineering regions near the coast. It also proves the correctness and validity of our method.

Kink, periodic and solitary solutions for coupled Benjamin–Bona–Mahony–KdV system