Transverse instability of gravity–capillary solitary waves

Kim, Boguk; Akylas, T. R.

doi:10.1007/s10665-006-9122-6

Cited by 21 publications

(32 citation statements)

References 14 publications

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“…If k y > k c given by (4.3) the weakly nonlinear stability analysis predicts stability (e.g. [14]), which is confirmed numerically. However, larger perturbations of the same transverse wavenumber destabilize the plane wave.…”

Section: Time-dependent Solutions and Stabilitysupporting

confidence: 56%

Transversally periodic solitary gravity–capillary waves

Milewski

Wang

2014

Proc. R. Soc. A.

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When both gravity and surface tension effects are present, surface solitary water waves are known to exist in both two-and three-dimensional infinitely deep fluids. We describe here solutions bridging these two cases: travelling waves which are localized in the propagation direction and periodic in the transverse direction. These transversally periodic gravity-capillary solitary waves are found to be of either elevation or depression type, tend to plane waves below a critical transverse period and tend to solitary lumps as the transverse period tends to infinity. The waves are found numerically in a Hamiltonian system for water waves simplified by a cubic truncation of the Dirichlet-to-Neumann operator. This approximation has been proved to be very accurate for both two-and three-dimensional computations of fully localized gravity-capillary solitary waves. The stability properties of these waves are then investigated via the time evolution of perturbed wave profiles.

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Section: Time-dependent Solutions and Stabilitysupporting

confidence: 56%

Transversally periodic solitary gravity–capillary waves

Milewski

Wang

2014

Proc. R. Soc. A.

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“…First, it is known that both depression and elevation plane solitary waves (1D waves extended in the second dimension) are linearly unstable with respect to sufficiently long small perturbations in the transverse direction. This has been shown both within an NLS approximation Rypdal & Rasmussen (1988) and using arguments based on linearization of the full equations Kim & Akylas (2007). Second, in 2D the underlying focussing NLS equation (2.14) is well known to exhibit a finite-time focussing blowup called wave collapse when, in an unbounded setting, the initial conserved energy E = |∇A| 2 − 1 2 |A| 4 , is negative (see Zakharov (1972) and Sulem & Sulem (1999)).…”

Section: Stability Focussing and Wave Collapsementioning

confidence: 96%

“…In 2D, there are far fewer studies. The transverse instability of line solitary waves (solitary waves of the 1D problem trivially extended in the transverse variable) has been considered by Kim & Akylas (2007) and others. Line solitary waves are unstable to transverse perturbations of sufficiently long wavelength.…”

Section: Introductionmentioning

confidence: 99%

Dynamics of gravity–capillary solitary waves in deep water

Wang

Milewski

2012

J. Fluid Mech.

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The dynamics of solitary gravity-capillary water waves propagating on the surface of a three-dimensional fluid domain is studied numerically. In order to accurately compute complex time dependent solutions, we simplify the full potential flow problem by taking a cubic truncation of the scaled Dirichlet-to-Neumann operator for the normal velocity on the free surface. This approximation agrees remarkably well with the full equations for the bifurcation curves, wave profiles and the dynamics of solitary waves for a twodimensional fluid domain. Fully localised solitary waves are then computed in the threedimensional problem and the stability and interaction of both line and localized solitary waves are investigated via numerical time integration of the equations. The solitary wave branches are indexed by their finite energy at small amplitude, and the dynamics of the solitary waves is complex involving nonlinear focussing of wave packets, quasi-elastic collisions, and the generation of propagating, spatially localised, time-periodic structures (breathers).

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“…In an attempt to fill this gap, Kim & Akylas (2007) examined the stability of plane solitary waves of depression to long-wavelength transverse perturbations. Depression solitary waves, while stable to longitudinal perturbations (Calvo & Akylas 2002), turn out to be transversely unstable, suggesting that, similarly to shallow-water lumps, lumps of the wave packet type may arise from the transverse instability of plane solitary waves.…”

Section: Introductionmentioning

confidence: 99%

On the stability of lumps and wave collapse in water waves

Akylas

Cho

2008

Phil. Trans. R. Soc. A.

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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.

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Transverse instability of gravity–capillary solitary waves

Cited by 21 publications

References 14 publications

Transversally periodic solitary gravity–capillary waves

Transversally periodic solitary gravity–capillary waves

Dynamics of gravity–capillary solitary waves in deep water

On the stability of lumps and wave collapse in water waves

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