Applicable symbolic computations on dynamics of small-amplitude long waves and Davey–Stewartson equations in finite water depth

•Analytical methods for wave problem are applied to study the surface gravity waves.•New solitary wave solutions of the Davey–Stewartson equations are obtained and presented.•The conservation laws and stability to the model system are derived.•Nonlinear dispersion relations and their components are discussed. Some classes of nonlinear partial differential equations can be reduced to more tractable single nonlinear equations via the lowest order of the perturbed reductive technique. The nonlinear and dispersive waves of the shallow-water model are investigated throughout a finite depth of fluid under the influence of surface tension and gravitational force in an attempt to derive the Davey–Stewartson equations (DSEs). Dispersion properties of the model and conservation laws of the DSEs are studied. We apply the Painlevé analysis to investigate the integrability of the DSEs and to construct the Bäcklund transformation via the truncation Painlevé expansion. By employing the Bäcklund transformation, the Hamiltonian approach and the (G′/G)-expansion method to the DSEs, new traveling solitary and kink wave solutions are obtained. It is revealed that the amplitudes of waves decrease with increasing Ursell parameter. The trend of the wave profile does not change with time. In addition, through the Hamiltonian approach, it is found that the amplitude of the waves increases with increasing energy constant. Furthermore, the phase portrait method is applied to the resulting nonlinear first-order differential equations of the DS model to reveal its stability.