scholarly journals Super compact equation for water waves

2017 ◽  
Vol 828 ◽  
pp. 661-679 ◽  
Author(s):  
A. I. Dyachenko ◽  
D. I. Kachulin ◽  
V. E. Zakharov
Keyword(s):  
Wave Equation ◽  
Water Waves ◽  
Wave Breaking ◽  
Water Wave ◽  
Rogue Waves ◽  
Resonant Interaction ◽  
Zakharov Equation ◽  
Advection Term ◽  
New Equation ◽  
The Many

Mathematicians and physicists have long been interested in the subject of water waves. The problems formulated in this subject can be considered fundamental, but many questions remain unanswered. For instance, a satisfactory analytic theory of such a common and important phenomenon as wave breaking has yet to be developed. Our knowledge of the formation of rogue waves is also fairly poor despite the many efforts devoted to this subject. One of the most important tasks of the theory of water waves is the construction of simplified mathematical models that are applicable to the description of these complex events under the assumption of weak nonlinearity. The Zakharov equation, as well as the nonlinear Schrödinger equation (NLSE) and the Dysthe equation (which are actually its simplifications), are among them. In this article, we derive a new modification of the Zakharov equation based on the assumption of unidirectionality (the assumption that all waves propagate in the same direction). To derive the new equation, we use the Hamiltonian form of the Euler equation for an ideal fluid and perform a very specific canonical transformation. This transformation is possible due to the ‘miraculous’ cancellation of the non-trivial four-wave resonant interaction in the one-dimensional wave field. The obtained equation is remarkably simple. We call the equation the ‘super compact water wave equation’. This equation includes a nonlinear wave term (à la NLSE) together with an advection term that can describe the initial stage of wave breaking. The NLSE and the Dysthe equations (DystheProc. R. Soc. Lond.A, vol. 369, 1979, pp. 105–114) can be easily derived from the super compact equation. This equation is also suitable for analytical studies as well as for numerical simulation. Moreover, this equation also allows one to derive a spatial version of the water wave equation that describes experiments in flumes and canals.

Numerical modelling of water-wave evolution based on the Zakharov equation

2001 ◽  
Vol 449 ◽  
pp. 341-371 ◽  
Author(s):  
SERGEI YU. ANNENKOV ◽  
VICTOR I. SHRIRA
Keyword(s):  
Numerical Modelling ◽  
Canonical Transformation ◽  
Water Waves ◽  
Water Wave ◽  
Numerical Models ◽  
Broad Band ◽  
Normal Modes ◽  
Physical Space ◽  
Zakharov Equation ◽  
Wave Evolution

We develop a new approach to numerical modelling of water-wave evolution based on the Zakharov integrodifferential equation and outline its areas of application.The Zakharov equation is known to follow from the exact equations of potential water waves by the symmetry-preserving truncation at a certain order in wave steepness. This equation, being formulated in terms of nonlinear normal variables, has long been recognized as an indispensable tool for theoretical analysis of surface wave dynamics. However, its potential as the basis for the numerical modelling of wave evolution has not been adequately explored. We partly fill this gap by presenting a new algorithm for the numerical simulation of the evolution of surface waves, based on the Hamiltonian form of the Zakharov equation taking account of quintet interactions. Time integration is performed either by a symplectic scheme, devised as a canonical transformation of a given order on a timestep, or by the conventional Runge–Kutta algorithm. In the latter case, non-conservative effects, small enough to preserve the Hamiltonian structure of the equation to the required order, can be taken into account. The bulky coefficients of the equation are computed only once, by a preprocessing routine, and stored in a convenient way in order to make the subsequent operations vectorized.The advantages of the present method over conventional numerical models are most apparent when the triplet interactions are not important. Then, due to the removal of non-resonant interactions by means of a canonical transformation, there are incomparably fewer interactions to consider and the integration can be carried out on the slow time scale (O(ε2), where ε is a small parameter characterizing wave slope), leading to a substantial gain in computational efficiency. For instance, a simulation of the long-term evolution of 103 normal modes requires only moderate computational resources; a corresponding simulation in physical space would involve millions of degrees of freedom and much smaller integration timestep.A number of examples aimed at problems of independent physical interest, where the use of other existing methods would have been difficult or impossible, illustrates various aspects of the implementation of the approach. The specific problems include establishing the range of validity of the deterministic description of water wave evolution, the emergence of sporadic horseshoe patterns on the water surface, and the study of the coupled evolution of a steep wave and low-intensity broad-band noise.

scholarly journals Hybrid waves consisting of the solitons, breathers and lumps for a (2+1)-dimensional extended shallow water wave equation

2021 ◽  
Author(s):  
Gao-Fu Deng ◽  
Yi-Tian Gao ◽  
Xin Yu ◽  
Cui-Cui Ding ◽  
Ting-Ting Jia ◽  
...  
Keyword(s):  
Wave Equation ◽  
Shallow Water ◽  
Water Waves ◽  
Water Wave ◽  
Lump Solutions ◽  
Shallow Water Wave ◽  
Hybrid Solutions ◽  
The One ◽  
Hybrid Waves ◽  
Positive Integers

Abstract Shallow water waves are studied for the applications in hydraulic engineering and environmental engineering. In this paper, a (2+1)-dimensional extended shallow water wave equation is investigated. Hybrid solutions consisting of H -soliton, M -breather and J -lump solutions have been constructed via the modified Pfaffian technique, where H , M and J are the positive integers. One-breather solutions with a real function ϕ ( y ) are derived, where y is the scaled space variable, we notice that ϕ ( y ) influences the shapes of the background planes. Discussions on the hybrid waves consisting of one breather and one soliton indicate that the one breather is not affected by one soliton after interaction. One-lump solutions with ϕ ( y ) are obtained with the condition, where k 1 R and k 1 I are the real constants, we notice that the one lump consists of two low valleys and one high peak, as well as the amplitude and velocity keep invariant during its propagation. Hybrid waves consisting of the one lump and one soliton imply that the shape of the one soliton becomes periodic when ϕ ( y ) is changed from a linear function to a periodic function.

SPECTRAL FORMULATION OF THE TWO FLUID EULER EQUATIONS WITH A FREE INTERFACE AND LONG WAVE REDUCTIONS

2008 ◽  
Vol 06 (04) ◽  
pp. 323-348 ◽  
Author(s):  
M. J. ABLOWITZ ◽  
T. S. HAUT
Keyword(s):  
Wave Equation ◽  
Water Waves ◽  
Water Wave ◽  
Wave Equations ◽  
Fluid System ◽  
Free Interface ◽  
Long Wave ◽  
Petviashvili Equation ◽  
Versus Amplitude ◽  
Two Fluid

A nonlocal spectral formulation of classic water waves is derived, and its connection to the classic water wave equations and the Dirichlet–Neumann operator is explored. The nonlocal spectral formulation is also extended to a two-fluid system with a free interface, from which long wave asymptotic reductions are obtained. Of particular interest is an asymptotically distinguished (2 + 1)-dimensional generalization of the intermediate long wave equation, which includes the Kadomtsev–Petviashvili equation and the Benjamin–Ono equation as limiting cases. Lump-type solutions to this (2 + 1)-dimensional ILW equation are obtained, and the speed versus amplitude relationship is shown to be linear in the shallow, intermediate, and deep water regimes.

A new equation describing travelling water waves

2013 ◽  
Vol 717 ◽  
pp. 514-522 ◽  
Author(s):  
Katie Oliveras ◽  
Vishal Vasan
Keyword(s):  
Water Waves ◽  
Water Wave ◽  
Single Equation ◽  
Travelling Wave ◽  
Higher Order ◽  
Stokes Wave ◽  
Two Dimensional ◽  
Wave Surface ◽  
New Equation ◽  
New Formulation

AbstractA new single equation for the surface elevation of a travelling water wave in an incompressible, inviscid, irrotational fluid is derived. This new equation is derived without approximation from Euler’s equations, valid for both a one- and two-dimensional travelling-wave surface. We show that this new formulation can be used to efficiently derive higher-order Stokes-wave approximations, and pose that this new formulation provides a useful framework for further investigation of travelling water waves.

scholarly journals Camassa-Holm equation on shallow water wave equation

2017 ◽  
Vol 5 (12) ◽  
pp. 7758-7764
Author(s):  
Sh. Hajrulla, L. Bezati, F. Hoxha
Keyword(s):  
Cauchy Problem ◽  
Wave Equation ◽  
Weak Solution ◽  
Shallow Water ◽  
Water Waves ◽  
Water Wave ◽  
Kdv Equation ◽  
Shallow Water Wave ◽  
Holm Equation ◽  
The Cauchy Problem

In this paper we can consider the problem of week solutions for the general shallow water wave equation. In the first part of this paper, we deal to the well-known Kdv equation. We obtain the Camassa-Holm equation in particular. Both of them describe unidirectional shallow water waves equation. Moreover, all these equations have a bi-Hamiltonian structure, they are completely integrable, they have infinitely many conserved quantities. From a mathematical point of view the Camassa-Holm equation is well studied. In the second part of this paper, we obtain a global weak solution as a limit of approximation under the assumption  Some concepts related to high dimensional spaces are considered. Then the Cauchy problem is considered. It has an admissible weak solution  to the Cauchy problem for  Existence, uniqueness, and basic energy estimate on this approximate solution sequence are given in some lemmas. Finally, the main theorem and the proof is given

scholarly journals On the Multipeakon Dissipative Behavior of the Modified Coupled Camassa-Holm Model for Shallow Water System

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Zhixi Shen ◽  
Yujuan Wang ◽  
Hamid Reza Karimi ◽  
Yongduan Song
Keyword(s):  
Differential Equations ◽  
Energy Loss ◽  
Shallow Water ◽  
Water Waves ◽  
Wave Breaking ◽  
Water Wave ◽  
Water System ◽  
Shallow Water Waves ◽  
Two Component ◽  
Shallow Water System

This paper investigates the multipeakon dissipative behavior of the modified coupled two-component Camassa-Holm system arisen from shallow water waves moving. To tackle this problem, we convert the original partial differential equations into a set of new differential equations by using skillfully defined characteristic and variables. Such treatment allows for the construction of the multipeakon solutions for the system. The peakon-antipeakon collisions as well as the dissipative behavior (energy loss) after wave breaking are closely examined. The results obtained herein are deemed valuable for understanding the inherent dynamic behavior of shallow water wave breaking.

scholarly journals On certain properties of the compact Zakharov equation

2014 ◽  
Vol 748 ◽  
pp. 692-711 ◽  
Author(s):  
Francesco Fedele
Keyword(s):  
Water Waves ◽  
Wave Breaking ◽  
Modulational Instability ◽  
Type Equation ◽  
Multiple Scale ◽  
Homoclinic Orbits ◽  
Dynamical Behaviour ◽  
Wave Steepness ◽  
Zakharov Equation ◽  
Long Time

AbstractLong-time evolution of a weakly perturbed wavetrain near the modulational instability (MI) threshold is examined within the framework of the compact Zakharov equation for unidirectional deep-water waves (Dyachenko and Zakharov, JETP Lett., vol. 93, 2011, pp. 701–705). Multiple-scale solutions reveal that a perturbation to a slightly unstable uniform wavetrain of steepness $\mu $ slowly evolves according to a nonlinear Schrodinger equation (NLS). In particular, for small carrier wave steepness $\mu <\mu _{1}\approx 0.27$ the perturbation dynamics is of focusing type and the long-time behaviour is characterized by the Fermi–Pasta–Ulam recurrence, the signature of breather interactions. However, the amplitude of breathers and their likelihood of occurrence tend to diminish as $\mu $ increases while the Benjamin–Feir index (BFI) decreases and becomes nil at $\mu _{1}$. This indicates that homoclinic orbits persist only for small values of wave steepness $\mu \ll \mu _{1}$, in agreement with recent experimental and numerical observations of breathers. When the compact Zakharov equation is beyond its nominal range of validity, i.e. for $\mu >\mu _{1}$, predictions seem to foreshadow a dynamical trend to wave breaking. In particular, the perturbation dynamics becomes of defocusing type, and nonlinearities tend to stabilize a linearly unstable wavetrain as the Fermi–Pasta–Ulam recurrence is suppressed. At $\mu =\mu _{c}\approx 0.577$, subharmonic perturbations restabilize and superharmonic instability appears, possibly indicating that wave dynamical behaviour changes at large steepness, in qualitative agreement with the numerical simulations of Longuet-Higgins and Cokelet (Proc. R. Soc. Lond. A, vol. 364, 1978, pp. 1–28) for steep waves. Indeed, for $\mu >\mu _{c}$ a multiple-scale perturbation analysis reveals that a weak narrowband perturbation to a uniform wavetrain evolves in accord with a modified Korteweg–de Vries/Camassa–Holm type equation, again implying a possible mechanism conducive to wave breaking.

Extended F-Expansion Method for Solving the modified Korteweg-de Vries (mKdV) Equation

2020 ◽  
Vol 11 (1) ◽  
pp. 93-100
Author(s):  
Vina Apriliani ◽  
Ikhsan Maulidi ◽  
Budi Azhari
Keyword(s):  
Wave Equation ◽  
Exact Solutions ◽  
Internal Waves ◽  
Water Waves ◽  
Wave Equations ◽  
Elliptic Functions ◽  
Expansion Method ◽  
Mkdv Equation ◽  
Innovative Methods ◽  
De Vries

One of the phenomenon in marine science that is often encountered is the phenomenon of water waves. Waves that occur below the surface of seawater are called internal waves. One of the mathematical models that can represent solitary internal waves is the modified Korteweg-de Vries (mKdV) equation. Many methods can be used to construct the solution of the mKdV wave equation, one of which is the extended F-expansion method. The purpose of this study is to determine the solution of the mKdV wave equation using the extended F-expansion method. The result of solving the mKdV wave equation is the exact solutions. The exact solutions of the mKdV wave equation are expressed in the Jacobi elliptic functions, trigonometric functions, and hyperbolic functions. From this research, it is expected to be able to add insight and knowledge about the implementation of the innovative methods for solving wave equations. 

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