LAUSR

Abstract 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 Painleve analysis to investigate the integrability of the DSEs and to construct the Backlund transformation via the truncation Painleve expansion. By employing the Backlund 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.