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Modelling of wave motion and propagation characteristics of waves plays an important role in coastal, ocean and maritime engineering, and for most geographical areas, waves are the major source of environmental actions on beaches or on man-made fixed or floating structures. The travelling wave solutions are described by the hyperbolic functions, the trigonometric functions and the rational functions. In this paper, exact travelling wave solutions of Hirota equation and Hirota–Maccari system are successfully surveyed. Extended trial equation method and generalized Kudryashov method are applied to gain some exact travelling wave solutions of Hirota equation and Hirota–Maccari system. Firstly, we find some exact travelling wave solutions of Hirota equation and Hirota–Maccari system with the aid of extended trial equation method. Secondly, we obtain dark soliton solutions of Hirota equation and Hirota–Maccari system by means of generalized Kudryashov method. Finally, for certain parameter values, we plot two and three dimensional graphics of imaginary and real values of some solutions that we obtained by using these methods. [ABSTRACT FROM AUTHOR]