The Green-Naghdi equations are a shallow water waves model which play important roles in nonlinear wave fields. By using the trial equation method and Complete discrimination system for polynomial we obtained classification of travelling patterns. Among those patterns, new singular patterns double periodic first time. And draw graphs help us to understand dynamics behaviors intuitionally.

this paper describes the effects of induced magnetic field and partial slip on the peristaltic flow of a couple stress fluids in an asymmetric channel. the two dimensional equation of couple stress fluid are simplified by making the assumptions of long wave length and low reynolds number. the exact solutions of reduced momentum equation and magnetic force function have been computed in wave fra...

We extend the formalism of statistical mechanics developed for the two-dimensional (2D) Euler equation to the case of shallow water system. Relaxation equations towards the maximum entropy state are proposed, which provide a parametrization of subgrid-scale eddies in 2D compressible turbulence.

It is known that the governing equations for the evolution of the envelope of weakly nonlinear capillary-gravity wave packets, the so-called Davey-Stewartson equation, admit dromion solutions in the limit of shallow water and strong surface tension, i.e., when kh≪1 and the Bond number (Bo) ≡T/ρgh^{2}>1/3 (where T is the surface tension coefficient and h the water depth). Here we show that capil...

Both physical oceanographic processes and marine geological features in the continental shelf can cause the medium properties to have lateral heterogeneity, so horizontal refraction of sound can occur and produce significant three-dimensional (3-D) sound propagation effects. The long-term goals of this project are targeted on understanding the 3-D acoustic effects caused by the environmental fa...

The generation of an undular bore in the vicinity of a wave-breaking point is considered for the integrable Kaup–Boussinesq (KB) shallow water system. In the framework of the Whitham modulation theory, an analytic solution of the Gurevich–Pitaevskii type of problem for a generic “cubic” breaking regime is obtained using a generalized hodograph transform, and a further reduction to a linear Eule...

The weakor wave-turbulence problem consists of finding statistical states of a large number of interacting waves. These states are obtained by forcing and dissipating a conservative dispersive wave equation at disparate scales to model physical forcing and dissipation, and by predicting the spectrum, often as a Kolmogorov-like power law, at intermediate scales. The mechanism for energy transfer...

A ( $$3+1$$ )-dimensional generalized shallow water waves equation is investigated with different methods. Based on symbolic computation and Hirota bilinear form, N-soliton solutions are constructed. In the process of degeneration solutions, T-breathers derived by taking complexication method. Then rogue will emerge during breathers parameter limit Through full N-soliton, M-lump based long-wave...