Wave Instabilities

Benney, D. J.; Roskes, G. J.

doi:10.1002/sapm1969484377

Cited by 426 publications

(290 citation statements)

References 5 publications

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“…Among many other physical situations, the NLS equation arises as an approximate model of the evolution of a nearly monochromatic wave of small amplitude in pulse propagation along optical fibers (αβ > 0) [16], in gravity waves on deep water (αβ < 0) [3,11], and in Langmuir waves in a plasma (αβ > 0) [19]. As a description of a Bose-Einstein condensate, the NLS is known as the GrossPitaevskii equation (αβ > 0) [20,14].…”

Section: Introductionmentioning

confidence: 99%

Stability of plane-wave solutions of a dissipative generalization of the nonlinear Schrödinger equation

Carter

Contreras

2008

Physica D: Nonlinear Phenomena

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The modulational instability of perturbed plane-wave solutions of the cubic nonlinear Schrödinger (NLS) equation is examined in the presence of three forms of dissipation. We present three families of decreasing-in-magnitude plane-wave solutions to this dissipative NLS equation. We establish that all such solutions that have no spatial dependence are linearly stable, though some perturbations may grow a finite amount. Further, we establish that all such solutions that have spatial dependence are linearly unstable if a certain form of dissipation is present.

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Section: Introductionmentioning

confidence: 99%

Stability of plane-wave solutions of a dissipative generalization of the nonlinear Schrödinger equation

Carter

Contreras

2008

Physica D: Nonlinear Phenomena

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“…Yuen and Lake (1975), in turn, derived the NLS equation on the basis of the averaged Lagrangian method. Benney and Roskes (1969) extended those two-dimensional theories to the case of three-dimensional wave perturbations in a finite-depth fluid and obtained equations that are now known as the Davey-Stewartson equations. In this particular case, the equation proves the existence of the transverse instability of a plane wave, which is much stronger than a longitudinal one.…”

Section: Introductionmentioning

confidence: 99%

Lagrange form of the nonlinear Schrödinger equation for low-vorticity waves in deep water

Abrashkin

Pelinovsky

2017

Nonlin. Processes Geophys.

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Abstract. The nonlinear Schrödinger (NLS) equation describing the propagation of weakly rotational wave packets in an infinitely deep fluid in Lagrangian coordinates has been derived. The vorticity is assumed to be an arbitrary function of Lagrangian coordinates and quadratic in the small parameter proportional to the wave steepness. The vorticity effects manifest themselves in a shift of the wave number in the carrier wave and in variation in the coefficient multiplying the nonlinear term. In the case of vorticity dependence on the vertical Lagrangian coordinate only (Gouyon waves), the shift of the wave number and the respective coefficient are constant. When the vorticity is dependent on both Lagrangian coordinates, the shift of the wave number is horizontally inhomogeneous. There are special cases (e.g., Gerstner waves) in which the vorticity is proportional to the squared wave amplitude and nonlinearity disappears, thus making the equations for wave packet dynamics linear. It is shown that the NLS solution for weakly rotational waves in the Eulerian variables may be obtained from the Lagrangian solution by simply changing the horizontal coordinates.

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“…It is well-known that gravity waves in infinite depth are long-wave unstable and suffer a side-band instability first identified both theoretically and experimentally by Benjamin & Feir (1967); the extension to finite depth fluid was provided by Benney & Roskes (1969). Superharmonic perturbations were investigated by Longuet-Higgins (1978), who found that the waves are stable if their amplitude does not exceed a critical value.…”

Section: Introductionmentioning

confidence: 99%

The stability of capillary waves on fluid sheets

Blyth

Părău

2016

J. Fluid Mech.

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The linear stability of finite amplitude capillary waves on inviscid sheets of fluid is investigated. A method similar to that recently used by Tiron & Choi (2012) to determine the stability of Crapper waves on fluid of infinite depth is developed by extending the conformal mapping technique of Dyachenko et al. (1996a) to a form capable of capturing general periodic waves on both the upper and the lower surface of the sheet, including the symmetric and antisymmetric waves studied by Kinnersley (1976). The primary, surprising result is that both symmetric and antisymmetric Kinnersley waves are unstable to small superharmonic disturbances. The waves are also unstable to subharmonic perturbations. Growth rates are computed for a range of steady waves in the Kinnersley family, and also waves found along the bifurcation branches identified by Blyth & Vanden-Broeck (2004). The instability results are corroborated by time integration of the fully nonlinear unsteady equations. Evidence is presented for superharmonic instability of nonlinear waves via a collision of eigenvalues on the imaginary axis which appear to have the same Krein signature.

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Wave Instabilities

Abstract: A uniform train of periodic waves may be unstable to large scale variations so that some incoherence can develop. The analysis is presented for the case of gravity waves, and for a class of interaction problems typical of many physical systems.

Cited by 426 publications

References 5 publications

Stability of plane-wave solutions of a dissipative generalization of the nonlinear Schrödinger equation

Stability of plane-wave solutions of a dissipative generalization of the nonlinear Schrödinger equation

Lagrange form of the nonlinear Schrödinger equation for low-vorticity waves in deep water

The stability of capillary waves on fluid sheets

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