Folded localized excitations in the (2+1)-dimensional modified dispersive water-wave system

By using a mapping approach and a linear variable separation approach, a new family of solitary wave solutions with arbitrary functions for the （2＋1）-dimensional modified dispersive water-wave system （MDWW） is derived. Based on the derived solutions and using some multi-valued functions, we obtain some novel folded localized excitations of the system.

New Exact Solutions and Evolution of Solitons in Background Waves for(2+1)-Dimensional Modified Dispersive Water-Wave System

With the help of the conditional similarity reduction method, some new exact solutions of the （2＋1）- dimensional modified dispersive water-wave system （MDWW） are obtained. Based on the derived solution, we investigate the evolution of solitons in the background waves.

Quasi-Periodic Structures Based on Symmetrical Lucas Function of (2+1)-Dimensional Modified Dispersive Water-Wave System

By introducing the Lucas-Riccati method and a linear variable separation method, new variable separation solutions with arbitrary functions are derived for a （2＋1）-dimensional modified dispersive water-wave system. The main idea of this method is to express the solutions of this system as polynomials in the solution of the Riecati equation that the symmetrical Lucas functions satisfy. From the variable separation sohition and by selecting appropriate functions, some novel Jacobian elliptic wave structure with variable modulus and their interactions with dromions and peakons are investigated.

Higher-Order Bragg Resonance in Gravity Surface Waves over Periodic Bottoms

A calculation method based on the Bloch theorem is developed for the gravity surface waves over the periodic bottoms of large undulations. The study shows the existence of comparable high-order bandgaps, which are demonstrated to result from the higher-order Bragg resonances, i.e. the resonant interactions between surface waves and the harmonic components of the fluctuating bottom. It is also shown that the band widths of the high-order gaps are quite sensitive to the amplitudes of high-order harmonics of the bottom.

Periodic folded waves for a (2+1)-dimensional modified dispersive water wave equation

A general solution, including three arbitrary functions, is obtained for a （2~l）-dimensional modified dispersive water-wave （MDWW） equation by means of the WTC truncation method. Introducing proper multiple valued functions and Jacobi elliptic functions in the seed solution, special types of periodic folded waves are derived. In the long wave limit these periodic folded wave patterns may degenerate into single localized folded solitary wave excitations. The interactions of the periodic folded waves and the degenerated single folded solitary waves are investigated graphically and found to be completely elastic.

Oblique Wave Scattering by An Undulating Porous Bottom in A Two-Layer Ice-Covered Fluid

The present study analyzes the reflection and transmission phenomenon of water-waves in a two-layer ice-covered system. The upper layer is covered by an ice-sheet, whereas the bottom of the lower layer is undulated and permeable. By using regular perturbation analysis and Fourier transform technique, the problem is solved and the first order reflection and transmission coefficients are determined. It is found that these coefficients depend on the shape as well as the permeability of the undulating bottom. Therefore, from the practical viewpoint, an undulating bottom topography is considered to determine all the aforesaid coefficients. The role of various system parameters, such as porosity, angle of incidence and ice parameters, are discussed to analyze the transformation of incident water wave energy from one layer to another layer. The outcomes are demonstrated in graphical forms.

Liquid-FEP-based U-tube triboelectric nanogenerator for harvesting water-wave energy

Harvesting ambient mechanical energy is a key technology for realizing self-powered electronics. With advantages of stability and durabilid, a liquid-solid-based triboelectric nanogenerator （TENG） has recently drawn much attention. However, the impacts of liquid properties on the TENG performance and the related working principle are still unclear. We assembled herein a U-tube TENG based on the liquid-solid mode and applied 11 liquids to study the effects of liquid properties on the TENG output performance. The results confirmed that the key factors influencing the output are polarity, dielectric constant, and affinity to fluorinated ethylene propylene （FEP）. Among the 11 liquids, the pure water-based U-tube TENG exhibited the best output with an open-circuit voltage （Voc） of 81.7 V and a short-circuit current （Isc） of 0.26 μA for the shaking mode （0.5 Hz）, which can further increase to 93.0 V and 0.48 μA, respectively, for the horizontal shifting mode （1.25 Hz）. The U-tube TENG can be utilized as a self-powered concentration sensor （component concentration or metal ion concentration） for an aqueous solution with an accuracy higher than 92%. Finally, an upgraded sandwich-like water-FEP U-tube TENG was applied to harvest water-wave energy, showing a high output with Voc of 350 V, Isc of 1.75 μA, and power density of 2.04 W/m3. We successfully lighted up 60 LEDs and powered a temperature-humidity meter. Given its high output performance, the water-FEP U-tube TENG is a very promising approach for harvesting water-wave energy for self-powered electronics.

Break-down criterion for the water-wave equation

We study the break-down mechanism of smooth solution for the gravity water-wave equation of infinite depth. It is proved that if the mean curvature κ of the free surface Σt, the trace（V, B） of the velocity at the free surface, and the outer normal derivative ？P/？n of the pressure P satisfy sup t∈[0,T]｜｜κ（t）｜｜Lp∩L^2＋∫0^T｜｜（▽V, ▽B）（t）｜｜L∞^6dt〈＋∞,inf （t,x,y）∈[0,T]×Σ_t- P/ n（t, x, y）≥c0,for some p 〉 2d and c_0〉 0, then the solution can be extended after t = T.

Symmetry Reduction of the(2+1)-Dimensional Modified Dispersive Water-Wave System

Using the standard truncated Painlev expansion, the residual symmetry of the(2+1)-dimensional modified dispersive water-wave system is localized in the properly prolonged system with the Lie point symmetry vector. Some different transformation invariances are derived by utilizing the obtained symmetries. The symmetries of the system are also derived through the Clarkson-Kruskal direct method, and several types of explicit reduction solutions relate to the trigonometric or the hyperbolic functions are obtained. Finally, some special solitons are depicted from one of the solutions.

Painlevé analysis, auto-Bäcklund transformations, bilinear forms and soliton solutions for a(2+1)-dimensional variable-coefficient modified dispersive water-wave system in fluid mechanics

In this paper,we investigate a(2+1)-dimensional variable-coefficient modified dispersive waterwave system in fluid mechanics.We prove the Painlevéintegrability for that system via the Painlevéanalysis.We find some auto-B?cklund transformations for that system via the truncated Painlevéexpansions.Bilinear forms and N-soliton solutions are constructed,where N is a positive integer.We discuss the inelastic interactions,elastic interactions and soliton resonances for the two solitons.We also graphically demonstrate that the velocities of the solitons are affected by the variable coefficient of that system.

On the local wellposedness of 3-D water wave problem with vorticity

In this article, we first present an equivalent formulation of the free boundary problem to 3-D incompressible Euler equations, then we announce our local wellposedness result concerning the free boundary problem in Sobolev space provided that there is no self-intersection point on the initial surface and under the stability assumption that $\frac{{\partial p}}{{\partial n}}(\xi )\left| {_{t = 0} } \right. \leqslant - 2c_0 < 0$ being restricted to the initial surface.