Three-Dimensional Steady Capillary-Gravity Waves

Hărăguş-Courcelle, Mariana; Kirchgässner, Klaus

doi:10.1007/978-3-642-56589-2_16

Cited by 6 publications

(9 citation statements)

References 26 publications

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“…A further degeneracy occurs when L additionally coincides with the k-axis (Figure 7(b)), so that θ 1 = ±π/2, θ 2 = 0. This special case, in which all eigenvalues are geometrically double, was studied in detail by Groves [10] and Haragus & Kirchgässner [13]. A degeneracy analogous to that shown in Figure 7(a) also occurs in cases II b and II c whenever L passes through one of the points of intersection of K 1 with C dr and K −1 with C dr ; a new geometrically double zero eigenvalue is created.…”

Section: Further Bifurcation Scenariosmentioning

confidence: 86%

“…At the linear level this symmetry manifests itself in the fact that all eigenvalues of the corresponding linear operator are geomtrically double, half of them corresponding to wave motions which are respectively symmetric and antisymmetric with respect to the symmetry operator. These special cases, and the symmetry reductions involved in them, have been discussed in detail by Groves & Mielke [11], Groves [10] and Haragus & Kirchgässner [13]. …”

Section: Theoremmentioning

confidence: 96%

“…., so that every pair of curves (C n , C m ), n < m intersect in precisely one point P nm (see Figure 9(b)). In the further special case θ 1 = ±π/2, θ 2 = 0 the mode n and mode −n eigenvalues for n = 0 again coincide to form geometrically double eigenvalues; this case was investigated in detail by Groves [10] and Haragus & Kirchgässner [13].…”

Section: Further Bifurcation Scenariosmentioning

confidence: 98%

“…Three-dimensional waves which are periodic in the direction of propagation and have qualitatively prescribed profiles in the perpendicular direction have been studied by Groves [10] and Haragus and Kirchgässner [13], who found waves whose profiles in the transverse direction resemble periodic, quasiperiodic, solitary or generalised solitary waves. These results are special, non-generic cases (θ 1 = 0, θ 2 = ±π/2 or θ 1 = ±π/2, θ 2 = 0) of the waves discussed in the present paper which require a specific analysis.…”

Section: Introductionmentioning

confidence: 99%

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A Bifurcation Theory for Three-Dimensional Oblique Travelling Gravity-Capillary Water Waves

Groves

Hărăguş

2003

Journal of Nonlinear Science

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This article presents a rigorous existence theory for small-amplitude three-dimensional travelling water waves. The hydrodynamic problem is formulated as an infinite-dimensional Hamiltonian system in which an arbitrary horizontal spatial direction is the time-like variable. Wave motions which are periodic in a second, different horizontal direction are detected using a centre-manifold reduction technique by which the problem is reduced to a locally equivalent Hamiltonian system with a finite number of degrees of freedom.A catalogue of bifurcation scenarios is compiled by means of a geometric argument based upon the classical dispersion relation for travelling water waves. Taking all parameters into account, one finds that this catalogue includes virtually any bifurcation or resonance known in Hamiltonian systems theory. Nonlinear bifurcation theory is carried out for a representative selection of bifurcation scenarios; solutions of the reduced Hamiltonian system are found by applying results from the well-developed theory of finite-dimensional Hamiltonian systems such as the Lyapunov centre theorem and the Birkhoff normal form.We find oblique line waves which depend only upon one spatial direction which is not aligned with the direction of wave propagation; the waves have periodic, solitary-wave or generalised solitary-wave profiles in this distinguished direction. Truly three-dimensional waves are also found which have periodic, solitary-wave or generalised solitary-wave profiles in one direction and are periodic in another. In particular, we recover doubly periodic waves with arbitrary fundamental domains and oblique versions of the results on threedimensional travelling waves already in the literature.

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Section: Further Bifurcation Scenariosmentioning

confidence: 86%

Section: Theoremmentioning

confidence: 96%

Section: Further Bifurcation Scenariosmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A Bifurcation Theory for Three-Dimensional Oblique Travelling Gravity-Capillary Water Waves

Groves

Hărăguş

2003

Journal of Nonlinear Science

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“…More specifically, they construct waves with a periodic, quasiperiodic, or generalized solitary wave profile in the x direction. Waves that are periodic in the direction of propagation X and have a bounded profile in the transverse direction Z (that is, choosing θ 1 = ± π /2, θ 2 = 0), were studied in Groves and Haragus‐Courcelle and Kirchgässner . The authors found waves with a periodic, quasiperiodic, or generalized solitary wave profile in the z direction.…”

Section: Introductionmentioning

confidence: 99%

Three‐dimensional internal gravity‐capillary waves in finite depth

Nilsson

2019

Math Methods in App Sciences

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We consider three‐dimensional inviscid‐irrotational flow in a two‐layer fluid under the effects of gravity and surface tension, where the upper fluid is bounded above by a rigid lid and the lower fluid is bounded below by a flat bottom. We use a spatial dynamics approach and formulate the steady Euler equations as an infinite‐dimensional Hamiltonian system, where an unbounded spatial direction x is considered as a time‐like coordinate. In addition, we consider wave motions that are periodic in another direction z. By analyzing the dispersion relation, we detect several bifurcation scenarios, two of which we study further: a type of 00(is)(iκ0) resonance and a Hamiltonian Hopf bifurcation. The bifurcations are investigated by performing a center‐manifold reduction, which yields a finite‐dimensional Hamiltonian system. For this finite‐dimensional system, we establish the existence of periodic and homoclinic orbits, which correspond to, respectively, doubly periodic travelling waves and oblique travelling waves with a dark or bright solitary wave profile in the x direction. The former are obtained using a variational Lyapunov‐Schmidt reduction and the latter by first applying a normal form transformation and then studying the resulting canonical system of equations.

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Some explicit solutions of the three-dimensional Euler equations with a free surface

Martin

2021

Math. Ann.

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We present a family of radial solutions (given in Eulerian coordinates) to the three-dimensional Euler equations in a fluid domain with a free surface and having finite depth. The solutions that we find exhibit vertical structure and a non-constant vorticity vector. Moreover, the flows described by these solutions display a density that depends on the depth. While the velocity field and the pressure function corresponding to these solutions are given explicitly through (relatively) simple formulas, the free surface defining function is specified (in general) implicitly by a functional equation which is analysed by functional analytic methods. The elaborate nature of the latter functional equation becomes simpler when the density function has a particular form leading to an explicit formula of the free surface. We subject these solutions to a stability analysis by means of a Wentzel–Kramers–Brillouin (WKB) ansatz.

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Three-Dimensional Steady Capillary-Gravity Waves

Cited by 6 publications

References 26 publications

A Bifurcation Theory for Three-Dimensional Oblique Travelling Gravity-Capillary Water Waves

A Bifurcation Theory for Three-Dimensional Oblique Travelling Gravity-Capillary Water Waves

Three‐dimensional internal gravity‐capillary waves in finite depth

Some explicit solutions of the three-dimensional Euler equations with a free surface

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