Hydroelastic solitary waves in deep water

The problem of waves propagating on the surface of a two-dimensional ideal fluid of infinite depth bounded above by an elastic sheet is studied with asymptotic and numerical methods. We use a nonlinear elastic model that has been used to describe the dynamics of ice sheets. Particular attention is paid to forced and unforced dynamics of waves having near-minimum phase speed. For the unforced problem, we find that wavepacket solitary waves bifurcate from nonlinear periodic waves of minimum speed. When the problem is forced by a moving load, we find that, for small-amplitude forcing, steady responses are possible at all subcritical speeds, but for larger loads there is a transcritical range of forcing speeds for which there are no steady solutions. In unsteady computations, we find that if the problem is forced at a speed in this range, very large unsteady responses are obtained, and that when the forcing is released, a solitary wave is generated. These solitary waves appear stable, and can coexist within a sea of small-amplitude waves.

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Experimental observation of gravity–capillary solitary waves generated by a moving air suction

Journal of Fluid Mechanics ◽

10.1017/jfm.2016.639 ◽

2016 ◽

Vol 808 ◽

pp. 168-188 ◽

Cited By ~ 6

Author(s):

Beomchan Park ◽

Yeunwoo Cho

Keyword(s):

Experimental Observation ◽

Solitary Waves ◽

Deep Water ◽

Small Amplitude ◽

Three Dimensional ◽

Phase Speed ◽

Linear Phase ◽

Wave Patterns ◽

Air Blowing ◽

Linear And Nonlinear

Gravity–capillary solitary waves are generated by a moving ‘air-suction’ forcing instead of a moving ‘air-blowing’ forcing. The air-suction forcing moves horizontally over the surface of deep water with speeds close to the minimum linear phase speed $c_{min}=23~\text{cm}~\text{s}^{-1}$. Three different states are observed according to forcing speeds below $c_{min}$. At relatively low speeds below $c_{min}$, small-amplitude linear circular depressions are observed, and they move steadily ahead of and along with the moving forcing. As the forcing speed increases close to $c_{min}$, however, nonlinear three-dimensional (3-D) gravity–capillary solitary waves are observed, and they move steadily ahead of and along with the moving forcing. Finally, when the forcing speed is very close to $c_{min}$, oblique shedding phenomena of 3-D gravity–capillary solitary waves are observed ahead of the moving forcing. We found that all the linear and nonlinear wave patterns generated by the air-suction forcing correspond to those generated by the air-blowing forcing. The main difference is that 3-D gravity–capillary solitary waves are observed ‘ahead of’ the air-suction forcing whereas the same waves are observed ‘behind’ the air-blowing forcing.

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Multiple gravity–capillary wave forms near the minimum phase speed

Journal of Fluid Mechanics ◽

10.1017/s0022112098004200 ◽

1999 ◽

Vol 384 ◽

pp. 93-106

Author(s):

JØRGEN H. RASMUSSEN ◽

MICHAEL STIASSNIE

Keyword(s):

Periodic Solutions ◽

Solitary Waves ◽

Present Note ◽

Capillary Wave ◽

Phase Speed ◽

Type Equation ◽

Minimum Phase ◽

Rich Family ◽

Wave Forms ◽

Preliminary Study

The existence of solitary waves near the minimum phase speed for waves in the gravity–capillary regime triggered our search for additional wave forms. We show that the governing Schrödinger-type equation also has a rich family of periodic solutions, and a preliminary study of these solutions is the objective of the present note.

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Steady waves in a cold plasma

Journal of Plasma Physics ◽

10.1017/s0022377800018778 ◽

1996 ◽

Vol 55 (2) ◽

pp. 181-194 ◽

Cited By ~ 8

Author(s):

Andrej Il'Ichev

Keyword(s):

Solitary Waves ◽

Small Amplitude ◽

Cold Plasma ◽

Travelling Waves ◽

Collisionless Plasma ◽

System Of Equations ◽

Periodic Waves ◽

Induction Vector ◽

Full System ◽

Different Types

The possible crest forms of magnetoacoustic travelling waves of small amplitude satisfying the full system of equations describe the wave propagation in a cold quasineutral collisionless plasma are determined. For a certain range of the inclination angle of the magnetic induction vector we find solitary waves. We find also families of periodic waves of two different types, as well as quasiperiodic waves aiid generalized solitary waves with non-decreasing oscillations at infinity.

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Bifurcation Phenomena of Nonlinear Waves in a Generalized Zakharov-Kuznetsov Equation

Advances in Mathematical Physics ◽

10.1155/2013/812120 ◽

2013 ◽

Vol 2013 ◽

pp. 1-14

Author(s):

Yun Wu ◽

Zhengrong Liu

Keyword(s):

Solitary Waves ◽

Nonlinear Waves ◽

Blow Up ◽

Periodic Waves ◽

The Third ◽

Bifurcation Phenomena ◽

Kink Waves ◽

Fourth Kind

We study the bifurcation phenomena of nonlinear waves described by a generalized Zakharov-Kuznetsov equationut+au2+bu4ux+γuxxx+δuxyy=0. We reveal four kinds of interesting bifurcation phenomena. The first kind is that the low-kink waves can be bifurcated from the symmetric solitary waves, the 1-blow-up waves, the tall-kink waves, and the antisymmetric solitary waves. The second kind is that the 1-blow-up waves can be bifurcated from the periodic-blow-up waves, the symmetric solitary waves, and the 2-blow-up waves. The third kind is that the periodic-blow-up waves can be bifurcated from the symmetric periodic waves. The fourth kind is that the tall-kink waves can be bifurcated from the symmetric periodic waves.

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On periodic water-waves and their convergence to solitary waves in the long-wave limit

Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences ◽

10.1098/rsta.1981.0231 ◽

1981 ◽

Vol 303 (1481) ◽

pp. 633-669 ◽

Cited By ~ 59

Keyword(s):

Integral Equation ◽

Solitary Waves ◽

Water Waves ◽

Periodic Problem ◽

Existence Theory ◽

Periodic Waves ◽

Equation Formulation ◽

Long Wave ◽

Integral Equation Formulation ◽

Wave Limit

A detailed discussion of Nekrasov’s approach to the steady water-wave problems leads to a new integral equation formulation of the periodic problem. This development allows the adaptation of the methods of Amick & Toland (1981) to show the convergence of periodic waves to solitary waves in the long-wave limit. In addition, it is shown how the classical integral equation formulation due to Nekrasov leads, via the Maximum Principle, to new results about qualitative features of periodic waves for which there has long been a global existence theory (Krasovskii 1961, Keady & Norbury 1978).

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