Three-Dimensional Long Water-Wave Phenomena

Akylas, T. R.

doi:10.1146/annurev.fl.26.010194.001203

Cited by 48 publications

(32 citation statements)

References 47 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…(Note that h A /0 as c/N, as required by the well-known constraint of the KP equation; see Akylas (1994).) Imposing this solvability condition on the r.h.s.…”

Section: Exchange Of Stabilitymentioning

confidence: 98%

On the stability of lumps and wave collapse in water waves

Akylas

Cho

2008

Phil. Trans. R. Soc. A.

View full text Add to dashboard Cite

In the classical water-wave problem, fully localized nonlinear waves of permanent form, commonly referred to as lumps, are possible only if both gravity and surface tension are present. While much attention has been paid to shallow-water lumps, which are generalizations of Korteweg-de Vries solitary waves, the present study is concerned with a distinct class of gravity-capillary lumps recently found on water of finite or infinite depth. In the near linear limit, these lumps resemble locally confined wave packets with envelope and wave crests moving at the same speed, and they can be approximated in terms of a particular steady solution (ground state) of an elliptic equation system of the Benney-Roskes-Davey-Stewartson (BRDS) type, which governs the coupled evolution of the envelope along with the induced mean flow. According to the BRDS equations, however, initial conditions above a certain threshold develop a singularity in finite time, known as wave collapse, due to nonlinear focusing; the ground state, in fact, being exactly at the threshold for collapse suggests that the newly discovered lumps are unstable. In an effort to understand the role of this singularity in the dynamics of lumps, here we consider the fifth-order Kadomtsev-Petviashvili equation, a model for weakly nonlinear gravity-capillary waves on water of finite depth when the Bond number is close to one-third, which also admits lumps of the wave packet type. It is found that an exchange of stability occurs at a certain finite wave steepness, lumps being unstable below but stable above this critical value. As a result, a small-amplitude lump, which is linearly unstable and according to the BRDS equations would be prone to wave collapse, depending on the perturbation, either decays into dispersive waves or evolves into an oscillatory state near a finite-amplitude stable lump.

show abstract

“…(Note that h A /0 as c/N, as required by the well-known constraint of the KP equation; see Akylas (1994).) Imposing this solvability condition on the r.h.s.…”

Section: Exchange Of Stabilitymentioning

confidence: 98%

On the stability of lumps and wave collapse in water waves

Akylas

Cho

2008

Phil. Trans. R. Soc. A.

View full text Add to dashboard Cite

show abstract

“…The governing Eqs. (2)(3)(4)(5)(6) can be transformed into dimensionless form (omitting the carets) in terms of (8):…”

Section: System Of Equationsmentioning

confidence: 99%

“…three-dimensional ocean waves and patterns [2], including perhaps the most simple, non-trivial, short-crested wave field composed of two progressive waves differing in the direction of propagation only. Short-crested wave is more predominant in nature than the progressive and standing waves normally analyzed theoretically and tested experimentally in the flume, which arise from oblique reflection on vertical or sloping surfaces, differential refraction across the continental shelf and swell resulting from two simultaneous storm centres, and are more relevant to sediment transport [3] and remote sensing in the open ocean [4].…”

mentioning

confidence: 99%

The Patterns of Surface Capillary-gravity Short-crested Waves with Uniform Current Fields in Coastal Waters

Huang

2006

Acta Mech Mech Sinica

View full text Add to dashboard Cite

A fully three-dimensional surface gravitycapillary short-crested wave system is studied as two progressive wave-trains of equal amplitude and frequency, which are collinear with uniform currents and doubly-periodic in the horizontal plane, are propagating at an angle to each other. The first-and second-order asymptotic analytical solutions of the short-crested wave system are obtained via a perturbation expansion in a small parameter associated with the wave steepness, therefore depicting a series of typical three-dimensional wave patterns involving currents, shallow and deep water, and surface capillary waves, and comparing them with each other.

show abstract

“…However, we do note that blow-up might occur if certain conditions on m, hold. In fact, the GKP equation 4 Ž . 1 , instability of lump-type solitary waves occurs when m) and s1, 3 and numerical results and asymptotic and dimensional arguments suggest w x blow-up in finite time 8 .…”

Section: The Generalized Boussinesq Equationmentioning

confidence: 99%

Initial Time Layers and Kadomtsev–Petviashvili‐Type Equations

Ablowitz

Wang

1997

Stud Appl Math

View full text Add to dashboard Cite

Ž .Ž . The Kadomtsev᎐Petviashvili KP equation and generalizations GKP have temporal discontinuities at the initial instant of time. Motivated by the study of water waves, a generalized Boussinesq equation that contains the GKP equations as an ''outer'' limit is introduced. Within the context of matched asymptotic expansions the discontinuities are resolved. The linear system is analyzed in more detail and the limit process is rigorously established.

show abstract

Three-Dimensional Long Water-Wave Phenomena

Cited by 48 publications

References 47 publications

On the stability of lumps and wave collapse in water waves

On the stability of lumps and wave collapse in water waves

The Patterns of Surface Capillary-gravity Short-crested Waves with Uniform Current Fields in Coastal Waters

Initial Time Layers and Kadomtsev–Petviashvili‐Type Equations

Contact Info

Product

Resources

About