Symmetry and the bifurcation of capillary-gravity waves

Jones, Mark; Toland, John

doi:10.1007/bf00251412

Cited by 57 publications

(39 citation statements)

References 12 publications

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“…Beale [1], who also allowed for small surface tension under conditions of nonresonance. Later J. Reeder and M. Shinbrot reproduced these results [24], and M. Jones and J. Toland [13], [14] dealt specifically with the occurrence of resonance.…”

Section: For Any Spatial Period There Exist Nontrivial Periodic Travementioning

confidence: 91%

Traveling Two and Three Dimensional Capillary Gravity Water Waves

Craig¹,

Nicholls²

2000

SIAM J. Math. Anal.

147

144

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Abstract. The main results of this paper are existence theorems for traveling gravity and capillary gravity water waves in two dimensions, and capillary gravity water waves in three dimensions, for any periodic fundamental domain. This is a problem in bifurcation theory, yielding curves in the two dimensional case and bifurcation surfaces in the three dimensional case. In order to address the presence of resonances, the proof is based on a variational formulation and a topological argument, which is related to the resonant Lyapunov center theorem. 1. Introduction. Nonlinear periodic traveling waves on the free surface of an ideal fluid tend to form hexagonal patterns. This phenomenon is the focus of a number of recent papers on the subject of water waves, and it is the topic of the present article. In previous work, various approximations to the evolution equations for free surface waves are used, in particular the KP system by J. Hammack, N. Scheffner, and H. Segur [11], and J. Hammack, D. McCallister, N. Scheffner, and H. Segur [12], and alternatively with certain formal shallow water expansions of the Euler equations by P. Milewski and J.B. Keller [16]. A natural question is whether similar patterns can be shown to occur in solutions of the full Euler equations themselves. This is the focus of a series of papers by the present authors. In [18] and in [20] we report on hexagonal wave patterns and other phenomena in numerical computations of solutions, which are shown to satisfy spectral criteria for numerical convergence to solutions of Euler's equations. In the present article we describe rigorous existence results for periodic traveling wave solutions in free surfaces. The goal is to prove the existence of nontrivial traveling wave solutions to the water wave problem for gravity and capillary gravity waves in two and three dimensions. In two dimensions this is proven for both gravity and capillary gravity water waves, constituting a new and relatively straightforward approach to the theorems of T. Levi-Civita and D. Struik. In three dimensions we prove the existence of traveling capillary gravity water waves. However, the problem of gravity waves in three dimensions exhibits the phenomena of small divisors, and it remains open. The theorem that we prove is given below.

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Section: For Any Spatial Period There Exist Nontrivial Periodic Travementioning

confidence: 91%

Traveling Two and Three Dimensional Capillary Gravity Water Waves

Craig¹,

Nicholls²

2000

SIAM J. Math. Anal.

147

144

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“…Jones & Toland [38] introduced and studied a version of Nekrasov's equation for periodic gravity-capillary waves which has become the basis for a large number of analytical and numerical existence theories revealing a much more complex local and global bifurcation picture than that for pure gravity waves. A comprehensive survey of current results in this area is given by Okamoto & Shoji [53].…”

Section: Further Resultsmentioning

confidence: 99%

Steady Water Waves

Groves¹

2004

JNMP

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“…Therefore the general solutions on this sheet represent a symmetry-breaking (or period-multiplying) secondary bifurcation on the curves of special solutions. This is the hydroelastic analogue of Wilton ripples, a type of water wave which arises in the presence of surface tension [8,14,15]. In Wilton-ripple theory there are also two parameters, the wave speed and the surface tension coefficient (which measures surface elasticity).…”

Section: Bifurcation Picturementioning

confidence: 99%

Bifurcation and secondary bifurcation of heavy periodic hydroelastic travelling waves

Baldi¹,

Toland²

2010

Interfaces Free Bound.

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The existence question for two-dimensional symmetric steady waves travelling on the surface of a deep ocean beneath a heavy elastic membrane is analyzed as a problem in bifurcation theory. The behaviour of the two-dimensional cross-section of the membrane is modelled as a thin (unshearable), heavy, hyperelastic extensible rod, and the fluid beneath is supposed to be in steady two-dimensional irrotational motion under gravity. When the wavelength has been normalized to be 2π , and when gravity and the density of the undeformed membrane are prescribed, there are two free parameters in the problem: the speed of the wave and the drift velocity of the membrane.It is observed that the problem, when linearized about uniform horizontal flow, has at most two independent solutions for any values of the parameters. When the linearized problem has only one normalized solution, it is shown that the full nonlinear problem has a sheet of solutions consisting of a family of curves bifurcating from simple eigenvalues. Here one of the problem's parameters is used to index a family of bifurcation problems in which the other is the bifurcation parameter.When the linearized problem has two solutions, with wave numbers k and l such that max{k, l}/min{k, l} / ∈ Z, it is shown that there are three two-dimensional sheets of bifurcating solutions. One consists of "special" solutions with minimal period 2π/k; another consists of "special" solutions with minimal period 2π/ l; and the third, apart from those on the curves where it intersects the "special" sheets, consists of "general" solutions with minimal period 2π .The two sheets of "special" solutions are rather similar to those that occur when the linearized problem has only one solution. However, points where the first sheet or the second sheet intersects the third sheet are period-multiplying (or symmetry-breaking) secondary bifurcation points on primary branches of "special" solutions. This phenomenon is analogous to that of Wilton ripples, which arises in the classical water-wave problem when the surface tension has special values. In the case of Wilton ripples, the coefficient of surface tension and the wave speed are the problem's two parameters. In the present context, there are two speed parameters, meaning that the membrane elasticity does not need to be highly specified for this symmetry-breaking phenomenon to occur.

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Symmetry and the bifurcation of capillary-gravity waves

Cited by 57 publications

References 12 publications

Traveling Two and Three Dimensional Capillary Gravity Water Waves

Traveling Two and Three Dimensional Capillary Gravity Water Waves

Steady Water Waves

Bifurcation and secondary bifurcation of heavy periodic hydroelastic travelling waves

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