Solitary wave solutions of the higher-order evolution equations for two ordering parameters in the shallow water waves

By introducing the relation between the orders of the two independent small expansion parameters, the amplitude parameter α and the long wavelength parameter β, which are taken in the form β=0αn and α=0βm, with n and m specified to two particular cases, we derive some new higher order partial differential nonlinear evolution equations for unidirectional shallow water waves under the influence of gravity as well as surface tension. We then show that it is possible to construct one-solution which satisfy these estimates using the Hirota’s bilinear method. •We derive in this paper a new family of equations describing the hydrodynamic waves.•We propose soliton-type solutions to its new equations.•We also study the influence of surface tension.