Water Wave System Research Articles

On the 3-dimensional water waves system above a flat bottom

As a starting point of studying the long time behavior of the $3D$ water waves system in the flat bottom setting, in this paper, we try to improve the understanding of the Dirichlet-Neumann operator in this setting. As an application, we study the $3D$ gravity waves system and derive a new $L^2-L^\infty$ type energy estimate, which has a good structure in the $L^\infty-$type space. In our second paper, base on the results we obtained in this paper, we prove the global regularity of the $3D$ gravity waves system for suitably small initial data.

Application of the ITEM for the modified dispersive water-wave system

The aim of this paper is to introduce a novel study of obtaining an analytical solutions to the modified dispersive water-wave system. An analytical technique based on the improved \(\tan (\phi /2)\)-expansion method (ITEM) is extended to handle such a system. Description of the method is given and the obtained results reveal that the ITEM is a new significant method for exploring nonlinear partial differential models. By using this method, exact solutions including the hyperbolic function solution, traveling wave solution, soliton solution, rational function solution, and periodic wave solution of this system of equations have been obtained. Moreover, by using Matlab, some graphical simulations were done to see the behavior of these solutions.

Nonlinear Self-Adjointness, Conservation Laws and Soliton-Cnoidal Wave Interaction Solutions of (2+1)-Dimensional Modified Dispersive Water-Wave System**Supported by National Natural Science Foundation of China under Grant Nos. 11371293, 11505090, the Natural Science Foundation of Shaanxi Province under Grant No. 2014JM2-1009, Research Award Foundation for Outstanding Young Scientists of Shandong Province under Grant No. BS2015SF009 and the Science and Technology Innovation Foundation of

This paper mainly discusses the (2+1)-dimensional modified dispersive water-wave (MDWW) system which will be proved nonlinear self-adjointness. This property is applied to construct conservation laws corresponding to the symmetries of the system. Moreover, via the truncated Painlevé analysis and consistent tanh-function expansion (CTE) method, the soliton-cnoidal periodic wave interaction solutions and corresponding images will be eventually achieved.

On the Cauchy problem for a generalized two-component shallow water wave system with fractional higher-order inertia operators

In this paper, we mainly consider the Cauchy problem for a generalized two-component shallow water wave system with fractional higher-order inertia operators: $m=(1-\partial_x^2)^su, s>1$. By Littlewood-Paley theory and transport equation theory, we first establish the local well-posedness of the generalized b-equation with fractional higher-order inertia operators which is the subsystem of the generalized two-component water wave system. Then we prove the local well-posedness of the generalized two-component water wave system with fractional higher-order inertia operators. Next, we present the blow-up criteria for these systems. Moreover, we obtain some global existence results for these systems.

Hadamard well-posedness of the gravity water waves system

We consider the system of (pure) gravity water waves in any dimension and in a fluid domain with a general bottom geometry. The unique solvability of this problem was established by Alazard–Burq–Zuily [Invent. Math. 198(1) (2014) 71–163] at a low regularity level where the initial surface is [Formula: see text] in terms of Sobolev embeddings; this result allows the existence of free surfaces with unbounded curvature. Our result states that the solutions obtained in the above work depend continuously on initial data in the strong topology in which the solutions are constructed. This establishes a well-posedness result in the sense of Hadamard.

Global Analysis of a Model for Capillary Water Waves in Two Dimensions

AbstractIn this paper we prove a global regularity result for a quadratic quasilinear model associated to the water waves system with surface tension and no gravity in dimension 2 (the capillary waves system). The model we consider here retains most of the difficulties of the full capillary water waves system, including the delicate time resonance structure and modified scattering. It is slightly simpler, however, at the technical level, and our goal here is to present our method in this simplified situation. The analysis of the full system is presented in a preprint by the authors (2014). © 2016 Wiley Periodicals, Inc.

The Dynamics of Flat Surface Internal Geophysical Waves with Currents

A two-dimensional water wave system is examined consisting of two discrete incompressible fluid domains separated by a free common interface. In a geophysical context this is a model of an internal wave, formed at a pycnocline or thermocline in the ocean. The system is considered as being bounded at the bottom and top by a flatbed and wave-free surface respectively. A current profile with depth-dependent currents in each domain is considered. The Hamiltonian of the system is determined and expressed in terms of canonical wave-related variables. Limiting behaviour is examined and compared to that of other known models. The linearised equations as well as long-wave approximations are presented.

Topological water wave states in a one-dimensional structure.

Topological concepts have been introduced into electronic, photonic, and phononic systems, but have not been studied in surface-water-wave systems. Here we study a one-dimensional periodic resonant surface-water-wave system and demonstrate its topological transition. By selecting three different water depths, we can construct different types of water waves - shallow, intermediate and deep water waves. The periodic surface-water-wave system consists of an array of cylindrical water tanks connected with narrow water channels. As the width of connecting channel varies, the band diagram undergoes a topological transition which can be further characterized by Zak phase. This topological transition holds true for shallow, intermediate and deep water waves. However, the interface state at the boundary separating two topologically distinct arrays of water tanks can exhibit different bands for shallow, intermediate and deep water waves. Our work studies for the first time topological properties of water wave systems, and paves the way to potential management of water waves.

New Complex Hyperbolic Function Solutions for the (2+1)-Dimensional Dispersive Long Water–Wave System

In this paper, new algorithms called the “Modified exp(−Ω)-expansion function method” and “Improved Bernoulli sub-equation function method” have been proposed. The first algorithm is based on the exp(−Ω(ξ))-expansion method; the latter is based on the Bernoulli sub-Ordinary Differential Equation method. The methods proposed have been expressed comprehensively in this manuscript. The analytical solutions and application results are presented by drawing the two- and three-dimensional surfaces of solutions such as hyperbolic, complex, trigonometric and exponential solutions for the (2+1)-dimensional dispersive long water–wave system. Finally, a conclusion has been presented by mentioning the important discoveries in this manuscript.

Wave Turbulence on Water Surface

We overview the wave turbulence approach by example of one physical system: gravity waves on the surface of an infinitely deep fluid. In the theoretical part of our review, we derive the nonlinear Hamiltonian equations governing the water-wave system and describe the premises of the weak wave turbulence theory. We outline derivation of the wave-kinetic equation and the equation for the probability density function, and most important solutions to these equations, including the Kolmogorov-Zakharov spectra corresponding to a direct and an inverse turbulent cascades, as well as solutions for non-Gaussian wave fields corresponding to intermittency. We also discuss strong wave turbulence as well as coherent structures and their interaction with random waves. We describe numerical and laboratory experiments, and field observations of gravity wave turbulence, and compare their results with theoretical predictions.

An application of -expansion method to the dispersive water system

The dispersive water wave system is explored by the -expansion method. Many new solutions are obtained with the aid of this method. These solutions include hyperbolic function solutions, trigonometric function solutions and rational function solutions. These new solutions will be better help us to understand the physical phenomenon of the disper- sive water wave equation.

On the absence of splash singularities in the case of two-fluid interfaces

We show that so-called splash singularities cannot develop in the case of locally smooth solutions of the two-fluid interfaces in two dimensions. More precisely, we show that the scenario of formation of singularities discovered by Castro, Córdoba, Fefferman, Gancedo, and Gómez-Serrano in the case of the water waves system, in which the interface remains locally smooth but self-intersects in finite time, is completely prevented in the case of two-fluid interfaces with positive densities.

Well-posedness and analytic solutions of a two-component water wave equation

We first establish the local well-posedness for the Cauchy problem of a two-component water waves system in a critical nonhomogeneous Besov space. Then, we derive a continuation criterion for strong solutions to the system. Finally, we prove the existence of analytic solutions to the system.

Conditional Similarity Reduction Method and Complex Wave Excitations for a High-Dimensional Nonlinear System

Abstract With the help of the conditional similarity reduction method, a new family of complex wave solutions with q=lx + my + kt + Γ(x, y, t) for the (2+1)-dimensional modified dispersive water-wave (MDWW) system are obtained. Based on the derived solitary wave solution, some novel complex wave localised excitations are investigated.

Symmetry Reduction of the (2+1)-Dimensional Modified Dispersive Water-Wave System* *Supported by the National Natural Science Foundation of China under Grant No. 11447017 and the Natural Science Foundation of Zhejiang Province under Grant Nos. LY14A010005 and LQ13A010013

Using the standard truncated Painlevé 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.

Traveling wave solutions for nonlinear dispersive water-wave systems with time-dependent coefficients

In this article, the solitary wave and topological soliton solutions in the models that describe the propagation of surface water waves in a uniform channel are successfully constructed. The solitary wave ansatz is used to carry out these distinct solutions. The corresponding integrability criteria, also known as constraint conditions, naturally emerge from the analysis of these models.

Wave pattern induced by a moving atmospheric pressure disturbance

A moving low atmospheric pressure is a main feature of tropical cyclones, which can induce a system of forced water waves and is an important factor that cause water level rise during a storm. A numerical model based on the nonlinear shallow water equations is applied to study the forced waves caused by an atmospheric pressure disturbance moving with a constant velocity over water surface. The effects of the moving speed, the spatial scale and the central pressure drop of the pressure disturbance are discussed. The results show that the wave pattern caused by a moving low-pressure is highly related with its moving speed. The wave pattern undergoes a great change as the moving speed approaches the wave velocity in shallow water. When the moving speed is less than the wave velocity, the distribution of water surface elevation is nearly the same as that of the pressure disturbance, and the maximum of the water surface elevation is located at the center of pressure. When the moving speed is larger than the wave velocity, a triangle shaped wave pattern is formed with a depression occurs in front of the pressure center, and the maximum of the water surface elevation lags behind the center of pressure. As the moving speed increases, the maximum of the water surface elevation firstly increases and then decreases, which reaches a peak when the moving speed is close to the wave velocity. The maximum of water surface elevation is approximately in proportion to the central pressure drop, and slightly affected by the spatial scale of pressure disturbance. Both the central pressure drop and the spatial scale of the pressure disturbance do not significantly affect the forced wave pattern. However, a clear difference can be noticed on the ratio of the maximum water surface elevation in moving pressure situation to that in static situation, when the moving speed is close to the wave velocity. A pressure disturbance with smaller spatial scale and smaller central pressure drop will give a larger ratio when the moving speed is close to the wave velocity.

Well-posedness, blow-up phenomena and persistence properties for a two-component water wave system

We first establish the local well-posedness for the Cauchy problem of a two-component water waves system in nonhomogeneous Besov spaces using the Littlewood–Paley theory. Then, we derive three new blow-up results for strong solutions to the system. Finally, we present two persistence properties for strong solutions to the system.

Multi-solitons and Related Solutions for the Water-waves System

The main result of this work is the construction of multi-solitons solutions, that is, solutions that are time asymptotics to a sum of decoupling solitary waves for the full water-waves system with surface tension. Our approach uses the construction of a precise approximate solution that is controlled by using spectral information for each solitary wave and a bootstrap argument for the control of the remainder. For this stage, we need only few properties about the nonlinear Cauchy problem, namely, local well-posedness for very smooth data. We also use a similar construction to refine our previous result [Invent. Math., 184 (2011), pp. 257--388] about the nonlinear instability of one-dimensional solitary waves in the two-dimensional model: we prove the existence of semiglobal solutions that depend nontrivially of the transverse variable and that tend to the line solitary wave as time goes to infinity.

Hamiltonian approach to the modeling of internal geophysical waves with vorticity

We examine a simplified model of internal geophysical waves in a rotational 2-dimensional water-wave system, under the influence of Coriolis forces and with gravitationally induced waves. The system consists of a lower medium, bound underneath by an impermeable flat bed, and an upper lid. The two media have a free common interface. Both media have constant density and constant (non-zero) vorticity. By examining the governing equations of the system we calculate the Hamiltonian of the system in terms of its conjugate variables and perform a variable transformation to show that it has canonical Hamiltonian structure. We then linearize the system, determine the equations of motion of the linearized system and calculate the dispersion relation. Finally, limiting cases are examined to recover irrotational and single medium systems as well as an infinite two media system.