A simple criterion for transverse linear instability of nonlinear waves

Godey, Cyril

doi:10.1016/j.crma.2015.10.017

Cited by 6 publications

(6 citation statements)

References 12 publications

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“…has a solution of the form e σt u σ (z), where Re σ > 0 and u σ is bounded (see Section 2.1 for a precise definition). The following theorem, which is due to Godey [10], shows that the existence of a pair of simple purely imaginary eigenvalues of L = df [u ] implies the transverse linear instability of u . Theorem 1.3 Consider the differential equation…”

Section: Introductionmentioning

confidence: 99%

A Dimension-Breaking Phenomenon for Water Waves with Weak Surface Tension

Groves

Sun

Wahlén

2015

Arch Rational Mech Anal

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It is well known that the water-wave problem with weak surface tension has smallamplitude line solitary-wave solutions which to leading order are described by the nonlinear Schrödinger equation. The present paper contains an existence theory for three-dimensional periodically modulated solitary-wave solutions which have a solitary-wave profile in the direction of propagation and are periodic in the transverse direction; they emanate from the line solitary waves in a dimension-breaking bifurcation. In addition, it is shown that the line solitary waves are linearly unstable to long-wavelength transverse perturbations. The key to these results is a formulation of the water wave problem as an evolutionary system in which the transverse horizontal variable plays the role of time, a careful study of the purely imaginary spectrum of the operator obtained by linearising the evolutionary system at a line solitary wave, and an application of an infinite-dimensional version of the classical Lyapunov centre theorem.

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Section: Introductionmentioning

confidence: 99%

A Dimension-Breaking Phenomenon for Water Waves with Weak Surface Tension

Groves

Sun

Wahlén

2015

Arch Rational Mech Anal

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“…Finally, we demonstrate the transverse linear instability of the line soliton using the following general result due to Godey [5]. Theorem 5.1 (Godey) Consider the differential equation…”

Section: Transverse Linear Instabilitymentioning

confidence: 99%

“…Finally, we demonstrate the transverse linear instability of the line soliton using the following general result due to Godey [5]. Theorem 5.1 (Godey) Consider the differential equation Under these hypotheses equation (25) has a solution of the form e λt v λ (τ ), where v λ ∈ C 1 (R, X ) ∩ C(R, Z) is periodic, for each sufficiently small positive value of λ; its period tends to 2π/ω 0 as λ → 0.…”

Section: Transverse Linear Instabilitymentioning

confidence: 99%

Periodic solitons for the elliptic–elliptic focussing Davey–Stewartson equations

Groves

Sun

Wahlén

2016

Comptes Rendus. Mathématique

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We consider the elliptic-elliptic, focussing Davey-Stewartson equations, which have an explicit bright line soliton solution. The existence of a family of periodic solitons, which have the profile of the line soliton in the longitudinal spatial direction and are periodic in the transverse spatial direction, is established using dynamical systems arguments. We also show that the line soliton is linearly unstable with respect to perturbations in the transverse direction. RésuméSolitons périodiques du système de Davey-Stewartson elliptique-elliptique focalisant. Nous considérons leséquations de Davey-Stewartson focalisantes dans le cas elliptique-elliptique, lorsqu'elles possèdent une solution unidimensionnelle de type soliton. En utilisant des méthodes de la théorie des systèmes dynamiques, nous montrons l'existence d'une famille de solutions bidimensionnelles qui ont le profil d'un soliton dans la direction spatiale longitudinale et sont périodiques dans la direction spatiale transverse. Nous montronségalement que le soliton unidimensionnel est linéairement instable vis-à-vis des perturbations transverses.

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“…This is the key, and most challenging, part of our analysis. The transverse stability of periodic waves was mostly studied for simpler model equations obtained from the Euler equations (1.1)-(1.2) in different parameter regimes: the Kadomtsev-Petviashvili-I equation for the regime of large surface tension (α ∼ 1, β > 1/3) was considered in [21,22,34], the Davey-Stewartson system for the regime of weak surface tension ((α, β) close to the curve Γ) in [13], and a fifth order KP equation for the regime of critical surface tension (α ∼ 1, β ∼ 1/3) in [25]; see also the recent review paper [24]. All these results predict that gravity-capillary periodic waves are linearly transversely unstable.…”

Section: Introductionmentioning

confidence: 99%

“…For the transverse linear instability problem, we consider the linearization of this dynamical system at a two-dimensional periodic wave and apply a simple general instability criterion [13] adapted to the Euler equations in [20]. In Region I, we show that the periodic waves…”

Section: Introductionmentioning

confidence: 99%

Transverse dynamics of two-dimensional traveling periodic gravity-capillary water waves

Hărăguş¹,

Truong²,

Wahlén³

2022

Preprint

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We study the transverse dynamics of two-dimensional traveling periodic waves for the gravity-capillary water-wave problem. The governing equations are the Euler equations for the irrotational flow of an inviscid fluid layer with free surface under the forces of gravity and surface tension. We focus on two open sets of dimensionless parameters (α, β), where α and β are the inverse square of the Froude number and the Weber number, respectively. For each arbitrary but fixed pair (α, β) in one of these sets, two-dimensional traveling periodic waves bifurcate from the trivial constant flow. In one open set we find a one-parameter family of periodic waves, whereas in the other open set we find two geometrically distinct one-parameter families of periodic waves. Starting from a transverse spatial dynamics formulation of the governing equations, we investigate the transverse linear instability of these periodic waves and the induced dimension-breaking bifurcation. The two results share a common analysis of the purely imaginary spectrum of the linearization at a periodic wave. We apply a simple general criterion for the transverse linear instability problem and a Lyapunov center theorem for the dimension-breaking bifurcation. For parameters (α, β) in the open set where there is only one family of periodic waves, we prove that these waves are linearly transversely unstable. For the other open set, we show that the waves with larger wavenumber are transversely linearly unstable. We also identify an open subset of parameters for which both families of periodic waves are tranversely linearly unstable. For each of these transversely linearly unstable periodic waves, a dimension-breaking bifurcation occurs in which three-dimensional doubly periodic waves bifurcate from the two-dimensional periodic wave.

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A simple criterion for transverse linear instability of nonlinear waves

Cited by 6 publications

References 12 publications

A Dimension-Breaking Phenomenon for Water Waves with Weak Surface Tension

A Dimension-Breaking Phenomenon for Water Waves with Weak Surface Tension

Periodic solitons for the elliptic–elliptic focussing Davey–Stewartson equations

Transverse dynamics of two-dimensional traveling periodic gravity-capillary water waves

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