Transverse instability of the line solitary water-waves

Rousset, Frédéric; Tzvetkov, Nikolay

doi:10.1007/s00222-010-0290-7

Cited by 61 publications

(67 citation statements)

References 39 publications

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“…For instance, in order to control the time derivatives of κ γ , one has to include the time derivatives of the unknowns in the energy. This method has been used by [15], [10] to study the Water Waves Problem with surface tension. Time derivatives and space derivatives play a different role in this proof, and we use the notation @k P N, ζ pkq " pεB t q k ζ, and ψ pkq " pεB t q k ψ´εwpεB t q k ζ for time derivatives, and…”

Section: The Energy Spacementioning

confidence: 99%

The Cauchy problem on large time for the Water Waves equations with large topography variations

Mésognon-Gireau¹

2017

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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bstract This paper shows that the long time existence of solutions to the Water Waves equations remains true for large amplitude topography variations in presence of surface tension. More precisely, the dimensionless equations depend strongly on three parameters ε, µ, β measuring the amplitude of the waves, the shallowness and the amplitude of the bathymetric variations respectively. In [2], the local existence of solutions to this problem is proved on a time interval of size 1 maxpβ,εq and uniformly with respect to µ. In presence of large bathymetric variations (typically β " ε), the existence time is therefore considerably reduced. We remove here this restriction and prove the local existence on a time interval of size 1 ε under the constraint that the surface tension parameter must be at the same order as the shallowness parameter µ. We also show that the result of [5] dealing with large bathymetric variations for the Shallow Water equations can be viewed as a particular endpoint case of our result.

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Section: The Energy Spacementioning

confidence: 99%

The Cauchy problem on large time for the Water Waves equations with large topography variations

Mésognon-Gireau¹

2017

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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“…At the same time, they became popular in the analysis of the stability of nonlinear waves, both relying upon functional-analytic methods and integrability techniques. As a model equation for surface water waves, some of the obtained results were extended to the Euler equations describing the full hydrodynamic problem [5,12,29].Stability properties of traveling waves are quite different for the two versions of the KP equation. While both periodic and solitary waves are transversely unstable in the KP-I equation (e.g., see recent works [8,15,27,28] and the references therein), it is expected that they are transversely stable in the KP-II equation [1,16].…”

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confidence: 99%

Counting Unstable Eigenvalues in Hamiltonian Spectral Problems via Commuting Operators

Hărăguş

Jin

Pelinovsky

2017

Commun. Math. Phys.

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We present a general counting result for the unstable eigenvalues of linear operators of the form JL in which J and L are skew-and self-adjoint operators, respectively. Assuming that there exists a self-adjoint operator K such that the operators JL and JK commute, we prove that the number of unstable eigenvalues of JL is bounded by the number of nonpositive eigenvalues of K. As an application, we discuss the transverse stability of one-dimensional periodic traveling waves in the classical KP-II (Kadomtsev-Petviashvili) equation. We show that these one-dimensional periodic waves are transversely spectrally stable with respect to general two-dimensional bounded perturbations, including periodic and localized perturbations in either the longitudinal or the transverse direction, and that they are transversely linearly stable with respect to doubly periodic perturbations.where the subscripts denote partial derivatives with respect to the spatial variables (x, y) and the temporal variable t. This equation is referred to as the KP-II equation, where the index II stands for the version relevant to the case of negative transverse dispersion. The KP-I equation is obtained by replacing the positive sign in front of the term u yy by a negative sign, and it is relevant to the case of positive transverse dispersion. Both versions of the KP equation are two-dimensional extensions of the KdV equation u t + 6uu x + u xxx = 0, (1.2) that governs one-dimensional nonlinear waves in the longitudinal direction of the x axis. Just like the KdV equation, the KP-II and KP-I equations arise as particular models in the classical water-wave problem, in the cases of small and large surface tension, respectively.The KP equations quickly became very popular due to their integrability properties [26], including a rich family of exact solutions, a bi-Hamiltonian structure and the recursion operator, a countable set of conserved quantities and symmetries, as well as the inverse scattering transform techniques. At the same time, they became popular in the analysis of the stability of nonlinear waves, both relying upon functional-analytic methods and integrability techniques. As a model equation for surface water waves, some of the obtained results were extended to the Euler equations describing the full hydrodynamic problem [5,12,29].Stability properties of traveling waves are quite different for the two versions of the KP equation. While both periodic and solitary waves are transversely unstable in the KP-I equation (e.g., see recent works [8,15,27,28] and the references therein), it is expected that they are transversely stable in the KP-II equation [1,16]. Numerical evidences of these stability properties can be found for instance in [19,20]. For the case of solitary waves, the transverse nonlinear stability has been recently proved for periodic transverse perturbations in [23], and for fully localized perturbations in [22]. In contrast, there are few analytical results for periodic waves for which, in particular, the question of transverse non...

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“…One motivation of the present paper was the study of their transverse stability. The transverse instability of the line solitary wave for some two dimensional models such as the nonlinear Schrödinger equation (NLS), the Kadomtsev-Petviashvili equation (KP) and some general "abstract" Hamiltonian systems have been carried out extensively in [42,43,44,29,30].…”

Section: The Local Well-posedness Inmentioning

confidence: 99%

On the Cauchy problem for the Zakharov-Rubenchik/ Benney-Roskes system

Luong¹,

Mauser²,

Saut³

2018

Communications on Pure &Amp; Applied Analysis

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We address various issues concerning the Cauchy problem for the Zakharov-Rubenchik system, (known as the Benney-Roskes system in water waves theory) which models the interaction of short and long waves in many physical situations. Motivated by the transverse stability/instability of the one-dimensional solitary wave (line solitary), we study the Cauchy problem in the background of a line solitary wave.

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Transverse instability of the line solitary water-waves

Abstract: Abstract. We prove the linear and nonlinear instability of the line solitary water waves with respect to transverse perturbations.

Cited by 61 publications

References 39 publications

The Cauchy problem on large time for the Water Waves equations with large topography variations

The Cauchy problem on large time for the Water Waves equations with large topography variations

Counting Unstable Eigenvalues in Hamiltonian Spectral Problems via Commuting Operators

On the Cauchy problem for the Zakharov-Rubenchik/ Benney-Roskes system

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