Under the well-known bilinear method of Hirota, specific expression for N-soliton solutions (2+1)-dimensional generalized Caudrey–Dodd–Gibbon–Kotera–Sawada(gCDGKS) equation in fluid mechanics is given. By defining a novel restrictive condition on solutions, resonant Y-type and X-type soliton are generated. new proposed constraint, combined with velocity resonance module method, mixed solitons l...

The Benney-Luke equation (BL) is a model for the evolution of three-dimensional weakly nonlinear, long water waves of small amplitude. In this paper we propose a nearly conservative scheme for the numerical resolution of (BL). Moreover, it is known ([PQ99] and [Pau03]) that (BL) is linked to the Kadomtsev-Petviashvili equation for almost one-dimensional waves propagating in one direction. We st...

We derive a new completely integrable dispersive shallow water equation that is biHamiltonian and thus possesses an infinite number of conservation laws in involution. The equation is obtained by using an asymptotic expansion directly in the Hamiltonian for Euler’s equations in the shallow water regime. The soliton solution for this equation has a limiting form that has a discontinuity in the f...

Wetland dominated estuaries serve as one of the most productive natural ecosystems through their ecological, economic and cultural services, such as nursery grounds for fisheries, nutrient sequestration, and ecotourism. The ongoing deterioration of wetland ecosystems in many shallow estuaries raises concerns about the contributing erosive processes and their roles in restraining coastal restora...

We present the derivation of the discrete Euler–Lagrange equations for an inverse spectral element ocean model based on the shallow water equations. We show that the discrete Euler–Lagrange equations can be obtained from the continuous Euler–Lagrange equations by using a correct combination of the weak and the strong forms of derivatives in the Galerkin integrals, and by changing the order with...

Unlike a transonic flow in compressible aerodynamics, a ship cruising steadily in a channel at nearly the linearized long wave speed generates unsteady wave motion. In a channel of limited width, solitons are radiated periodically up-stream. Such waves may have been the cause of a fatal accident at the Port of Harwich, England, and were described by a witness “like the white cliffs of Dover”. A...

In the present study, the nonlinear Boussinesq type equation describe the bi-directional propagation of small amplitude long capillary–gravity waves on the surface of shallow water. By using the extended auxiliary equation method, we obtained some new soliton like solutions for the two-dimensional fourth-order nonlinear Boussinesq equation with constant coefficient. These solutions include symm...