Spatial Form of a Hamiltonian Dysthe Equation for Deep-Water Gravity Waves

Guyenne, Philippe; Kairzhan, Adilbek; Sulem, Catherine; Xu, Boyang

doi:10.3390/fluids6030103

Cited by 6 publications

(8 citation statements)

References 49 publications

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“…the reduced Hamiltonian) over time. Our methodology, introduced recently for two-and three-dimensional irrotational gravity waves (Craig et al 2021a;Guyenne et al 2021Guyenne et al , 2022, consists in performing a sequence of canonical transformations that involve a reduction to normal form (devoid of non-resonant triads) and use of a modulational ansatz together with a scale separation lemma. A novelty of our approach is a direct reconstruction of the surface variables from the wave envelope through inversion of the third-order normal form transformation.…”

Section: Discussionmentioning

confidence: 99%

“…which is its linearization along the Burgers flow. These equations were tested in Craig et al (2021a) and Guyenne et al (2021Guyenne et al ( , 2022 in the context of irrotational gravity waves on deep water. Due to the complexity of the formulas for ∂ η K (3) and ∂ ζ K (3) , we have checked a posteriori that the cohomology equation (3.17) is indeed satisfied.…”

Section: Third-order Birkhoff Normal Formmentioning

confidence: 99%

“…In the absence of vorticity (), the equation for simplifies to , or equivalently, to the inviscid Burgers equation for , as obtained by Craig & Sulem (2016), while satisfies which is its linearization along the Burgers flow. These equations were tested in Craig et al (2021 a ) and Guyenne et al (2021, 2022) in the context of irrotational gravity waves on deep water.…”

Section: Birkhoff Normal Form Transformationsmentioning

confidence: 99%

“…It has been observed that numerical solutions of the Dysthe model provide a better agreement with laboratory experiments than the NLS equation, and are able e.g. to emulate the asymmetry of propagating wave packets, a feature not captured by the NLS equation (Lo & Mei 1985; Guyenne et al 2021).…”

Section: Introductionmentioning

confidence: 99%

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A Hamiltonian Dysthe equation for deep-water gravity waves with constant vorticity

Guyenne

Kairzhan²,

Sulem³

2022

J. Fluid Mech.

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This paper is a study of the water wave problem in a two-dimensional domain of infinite depth in the presence of non-zero constant vorticity. A goal is to describe the effects of uniform shear flow on the modulation of weakly nonlinear quasi-monochromatic surface gravity waves. Starting from the Hamiltonian formulation of this problem and using techniques from Hamiltonian transformation theory, we derive a Hamiltonian Dysthe equation for the time evolution of the wave envelope. Consistent with previous studies, we observe that the uniform shear flow tends to enhance or weaken the modulational instability of Stokes waves depending on its direction and strength. Our method also provides a non-perturbative procedure to reconstruct the surface elevation from the wave envelope, based on the Birkhoff normal form transformation to eliminate all non-resonant triads. This model is tested against direct numerical simulations of the full Euler equations and against a related Dysthe equation derived recently by Curtis, Carter & Kalisch (J. Fluid Mech., vol. 855, 2018, pp. 322–350) in the context of constant vorticity. Very good agreement is found for a range of values of the vorticity.

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Section: Discussionmentioning

confidence: 99%

Section: Third-order Birkhoff Normal Formmentioning

confidence: 99%

Section: Birkhoff Normal Form Transformationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A Hamiltonian Dysthe equation for deep-water gravity waves with constant vorticity

Guyenne

Kairzhan²,

Sulem³

2022

J. Fluid Mech.

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“…In a subsequent paper [25], we derived a spatial form of this Hamiltonian Dysthe equation, better adapted to comparison with measurements along a wave tank in laboratory experiments, and we tested it against the classical results of Lo and Mei [31]. We mention that a Hamiltonian version of the Dysthe equation was derived by Fedele and Dutykh [21] but it is not obtained directly from the water wave problem.…”

Section: Introductionmentioning

confidence: 99%

Normal form transformations for modulated deep-water gravity waves

Guyenne¹,

Kairzhan²,

Sulem³

2023

Revista De La Unión Matemática Argentina

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Modulation theory is a well-known tool to describe the long-time evolution and stability of small-amplitude, oscillating solutions to dispersive nonlinear partial differential equations. There have been a number of approaches to deriving envelope equations for weakly nonlinear waves. Here we review a systematic method based on Hamiltonian transformation theory and averaging Hamiltonians. In the context of the modulation of two-or three-dimensional deep-water surface waves, this approach leads to a Dysthe equation that preserves the Hamiltonian character of the water wave problem. An explicit calculation of the third-order Birkhoff normal form that eliminates all non-resonant cubic terms yields a non-perturbative procedure for the reconstruction of the free surface. We also present new numerical simulations of this weakly nonlinear approximation using the version with exact linear dispersion. We compare them against computations from the full water wave system and find very good agreement.

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A Hamiltonian Dysthe equation for hydroelastic waves in a compressed ice sheet

Guyenne,

Kairzhan,

Sulem

2025

J. Fluid Mech.

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Nonlinear hydroelastic waves along a compressed ice sheet lying on top of a two-dimensional fluid of infinite depth are investigated. Based on a Hamiltonian formulation of this problem and by applying techniques from Hamiltonian perturbation theory, a Hamiltonian Dysthe equation is derived for the slowly varying envelope of modulated wavetrains. This derivation is further complicated here by the presence of cubic resonances for which a detailed analysis is given. A Birkhoff normal form transformation is introduced to eliminate non-resonant triads while accommodating resonant ones. It also provides a non-perturbative scheme to reconstruct the ice-sheet deformation from the wave envelope. Linear predictions on the modulational instability of Stokes waves in sea ice are established, and implications for the existence of solitary wave packets are discussed for a range of values of ice compression relative to ice bending. This Dysthe equation is solved numerically to test these predictions. Its numerical solutions are compared with direct simulations of the full Euler system, and very good agreement is observed.

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Spatial Form of a Hamiltonian Dysthe Equation for Deep-Water Gravity Waves

Cited by 6 publications

References 49 publications

A Hamiltonian Dysthe equation for deep-water gravity waves with constant vorticity

A Hamiltonian Dysthe equation for deep-water gravity waves with constant vorticity

Normal form transformations for modulated deep-water gravity waves

A Hamiltonian Dysthe equation for hydroelastic waves in a compressed ice sheet

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