Hamiltonian long-wave approximations to the water-wave problem

Craig, Walter; Groves, Mark D.

doi:10.1016/0165-2125(94)90003-5

Cited by 178 publications

(159 citation statements)

References 24 publications

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“…These two equations can be used to determine the wave height and the velocity potential on the free surface. From this system well-known asymptotic equations, in both shallow and deep water with surface tension included, are obtained and agree in the shallow water limit with the results in Craig & Groves (1994). Furthermore, computational techniques are developed which provide a framework for performing fully nonlinear water wave simulations.…”

Section: Introductionmentioning

confidence: 99%

“…We mention that in Craig & Groves (1994) the series expansion of the DirichletNeumann map was used to derive small-amplitude/long-wave equations for the Boussinesq and KP equations without surface tension.…”

Section: Two Dimensional Boussinesq Benney-luke and Kp Equationsmentioning

confidence: 99%

“…This formulation was used in Craig & Groves (1994) to find small-amplitude/long-wave approximations including the Boussinesq, Korteweg-deVries (KdV) and KadomtsevPetviashvili (KP) equations. In Craig & Nicholls (2000) this formulation was used to prove the existence of travelling periodic water waves.…”

Section: Introductionmentioning

confidence: 99%

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On a new non-local formulation of water waves

2006

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The classical equations of water waves are reformulated as a system of two equations, one of which is an explicit non-local equation, for the wave height and for the velocity potential evaluated on the free surface. Evaluation of the velocity potential as a function of the depth is not required in order to calculate the wave height and the velocity potential on the free surface. The non-local system yields integral relations related to mass and centre of mass, and is shown to reduce to known asymptotic limits in shallow and deep water. Included in these asymptotic reductions are the Boussinesq, Benney-Luke and nonlinear Schrödinger equations. Two-dimensional lumps with sufficient surface tension are obtained numerically. The extension of this non-local formulation to the case of a variable bottom is also presented.

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Section: Introductionmentioning

confidence: 99%

Section: Two Dimensional Boussinesq Benney-luke and Kp Equationsmentioning

confidence: 99%

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On a new non-local formulation of water waves

2006

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“…Such equations have been studied by many authors as long-wave approximations to the waterwave equations. See, for example, Benney [2] and Olver [13] for the nonlinearity F (u) = −u 3 , Zufiria [14] and Hunter & Scheurle [9] for the nonlinearity F (u, u x ) = −uu 2 x , and Craig & Groves [6] for the nonlinearity F (u, u x ) = uu 2 x − u 3 . Traveling wave solutions u(x, t) = ϕ(x + ct) of equation (1.1) are described by the stationary equation…”

Section: Introductionmentioning

confidence: 99%

Stability of solitary waves of a fifth-order water wave model

Levandosky

2007

Physica D: Nonlinear Phenomena

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We consider the stability of solitary waves of a class of 5th order KdV equations. It is known that their stability is determined by the second derivative of a function of the wave speed d(c). We perform a detailed investigation of the properties of this function, both analytically and numerically. For a class of homogeneous nonlinearities, we precisely determine the regions of wave speeds for which the solitary waves are stable or unstable.

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“…For the initial value problem, Boundary Perturbations applied to the formulation of Zakharov have been very fruitful in the derivation of long-wave approximations of the Euler equations (Craig, Sulem, & Sulem [15]; Craig & Groves [18]; Craig, Guyenne, Nicholls, & Sulem [22]; and Craig, Guyenne, & Kalisch [21]), and the examination of integrability properties of the Euler equations (Craig & Worfolk [27]; Craig & Groves [19]; and Craig [17]). …”

Section: Introductionmentioning

confidence: 99%

Boundary Perturbation Methods for Water Waves

Nicholls

2007

GAMM-Mitteilungen

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Key words Water waves, free-surface fluid flows, ideal fluid flows, boundary perturbation methods, spectral methods.The most successful equations for the modeling of ocean wave phenomena are the freesurface Euler equations. Their solutions accurately approximate a wide range of physical problems from open-ocean transport of pollutants, to the forces exerted upon oil platforms by rogue waves, to shoaling and breaking of waves in nearshore regions. These equations provide numerous challenges for theoreticians and practitioners alike as they couple the difficulties of a free boundary problem with the subtle balancing of nonlinearity and dispersion in the absence of dissipation. In this paper we give an overview of, what we term, "Boundary Perturbation" methods for the analysis and numerical simulation of this "water wave problem." Due to our own research interests this review is focused upon the numerical simulation of traveling water waves, however, the extensive literature on the initial value problem and additional theoretical developments are also briefly discussed.

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Hamiltonian long-wave approximations to the water-wave problem

Cited by 178 publications

References 24 publications

On a new non-local formulation of water waves

On a new non-local formulation of water waves

Stability of solitary waves of a fifth-order water wave model

Boundary Perturbation Methods for Water Waves

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