The patterns of traveling wave on shallow water modeled by Green-Naghdi equation

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 the Complete discrimination system for the polynomial we obtained the classification of travelling wave patterns. Among those patterns, new singular patterns and double periodic patterns are obtained in the first time. And we draw the graphs which help us to understand the dynamics behaviors of the Green-Naghdi model intuitionally.

Degeneration of Solitons for a the (3+1)-dimensional Generalized Nonlinear Evolution Equation for the Shallow Water Waves

A the (3+1)-dimensional generalized nonlinear evolution equation for the shallow water waves is investigated with different methods. Based on symbolic computation and Hirota bilinear form, Nsoliton solutions are constructed. In the process of degeneration of N-soliton solutions, T-breathers are derived by taking complexication method. Then rogue waves will emerge during the degeneration of breathers by taking the parameter limit method. Through full degeneration of N-soliton, M-lump solutions are derived based on long wave limit approach. In addition, we also find out that the partial degeneration of N-soliton process can generate the hybrid solutions composed of soliton, breather and lump.

Nonlinear dynamical study to time fractional Dullian–Gottwald–Holm model of shallow water waves

This paper studies the exact traveling wave solutions to the nonlinear Dullin–Gottwald–Holm model which has the application in shallow-water waves in which the fractional derivative is considered in the sense of conformable derivative. Diverse exact solutions in hyperbolic, trigonometric and plane wave forms are obtained using two integration norms. For this purpose [Formula: see text]-expansion method and reccati mapping techniques are used. The 3D plots and their corresponding contour graphs are also depicted. Being concise and straightforward, the calculations demonstrate the effectiveness and convenience of the method for solving other nonlinear partial differential equations.

New Exact Travelling Wave Solutions for Fractional Differential Equations in a Shallow Water Waves

In this article, the modified simple equation method is proposed to solve nonlinear space-time fractional differential equations. This method is applied to solve space-time fractional modified Benjamin-Bona-Mahony (mBBM) equation, the space-time fractional generalized reaction duffing model and the space-time fractional potential Kadomtsev-Petviashvili (pKP) equation. The solutions found are hyperbolic and trigonometric function solutions. Some of these solutions are new solutions that are not available in the literature.

Analytical Treatment of the Generalized Hirota-Satsuma-Ito Equation Arising in Shallow Water Wave

In the current study, an analytical treatment is studied starting from the 2 + 1 -dimensional generalized Hirota-Satsuma-Ito (HSI) equation. Based on the equation, we first establish the evolution equation and obtain rational function solutions by means of the bilinear form with the help of the Hirota bilinear operator. Then, by the suggested method, the periodic, cross-kink wave solutions are also obtained. Also, the semi-inverse variational principle (SIVP) will be utilized for the generalized HSI equation. Two major cases were investigated from two different techniques. Moreover, the improved tan χ ξ method on the generalized Hirota-Satsuma-Ito equation is probed. The 3D, density, and contour graphs illustrating some instances of got solutions have been demonstrated from a selection of some suitable parameters. The existing conditions are handled to discuss the available got solutions. The current work is extensively utilized to report plenty of attractive physical phenomena in the areas of shallow water waves and so on.

Seiches and the slide/seiche dynamics; subcritical and supercritical subaquous mass flows and their deposits. Examples from Swiss Lakes

Four historically documented large and potentially dangerous lacustrine waves in Swiss lakes show that these waves have been seiches (standing waves) triggered by sublacustrine slides; a statement which is in accordance with the experience of seismologists who see earthquakes triggering seiches in lakes. Nevertheless, large historical waves in Switzerland have recently been modeled as progressive shallow water waves (tsunamis), probably because the slide/seiche dynamics are not known, and experiments with subaquatic slides fail to generate seiches in test–flumes. It appears that these tests exhibit a small shear–energy/slide–energy ratio ε, if compared with the situation in lakes. These facts incite a shear–stress lemma that states that ε is the constituent factor for the slide/seiche coupling. The structure of the subaqueous mass flow deposit (MFD) in lakes Lucerne and Geneva suggests the occurrence of subcritical and of supercritical slide flows. The former would generate a contortite, a MFD with contorted bedding, the latter a debrite (mudclast conglomerate). Potential slide energy considerations are used for an estimation of the amplitudes of large seiches produced by subaquatic slides, a proceeding that yields partly similar and partly very different results, as compared with numerical tsunami simulations.