A comparative study of bi-directional Whitham systems

Dinvay, Evgueni; Dutykh, Denys; Kalisch, Henrik

doi:10.1016/j.apnum.2018.09.016

Cited by 28 publications

(64 citation statements)

References 32 publications

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“…Numerical simulations done in [10] show how insignificantly values of functional H differ from the corresponding energy levels of the full water problem. We also consider a system posed on R 2+1 of the following Whitham-Boussinesq type…”

Section: Introductionmentioning

confidence: 99%

“…In the long wave Boussinesq regime v coincides with the horizontal fluid velocity at the surface. The system (1.1) possesses a Hamiltonian structure [10]. To our knowledge, there are at least two conserved quantities associated with this system.…”

Section: Introductionmentioning

confidence: 99%

“…Equations (1.1) were firstly proposed and studied numerically in [10]. Later in [9] the first proof of local well-posedness based on an energy method and a compactness argument was given.…”

Section: Introductionmentioning

confidence: 99%

“…If one in addition formally discards the term η∂ x u in the system given in [15], then a new alternative system turns out to be locally well-posed and features wave breaking [16]. However, the latter does not belong to the class of Boussinesq-Whitham models since nonlinear non-dispersive terms have been neglected.We would like to pay special attention to a system that was not considered in [10] but was introduced by Duchêne, Israwi and Talhouk [11]. They modified the bi-layer Green-Naghdi model improving the frequency dispersion.…”

mentioning

confidence: 99%

“…And indeed, as noticed in [11], their modification gives more reliable results when it comes to large-frequency Kelvin-Helmholtz instabilities than other models of the Green-Naghdi type.On the contrary, System (1.1) has a couple of advantages compared with the modified Green-Naghdi model [11]. Firstly, it is derived, though not rigorously, from the Zakharov-Craig-Sulem formulation, and as a result one knows the relation between variables (η, v) and those describing the full potential fluid flow [10]. As to the modification discussed, it is presented in variables where the first one has the meaning of the surface elevation and so coincides with η.…”

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confidence: 99%

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On well-posedness of a dispersive system of the Whitham–Boussinesq type

Dinvay

2019

Applied Mathematics Letters

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We regard the Cauchy problem for a particular Whitham-Boussinesq system modelling surface waves of an inviscid incompressible fluid layer. We are interested in well-posedness at a very low level of regularity. We derive dispersive and Strichartz estimates, and implement them together with a fixed point argument to solve the problem locally. Hamiltonian conservation guarantees global well-posedness for small initial data in the one dimensional settings. 1In fact (1.1) can be regarded itself as a regularization of the system introduced by Hur and Pandey [15]. The latter was also investigated numerically in [10] and compared with other models of Whitham-Boussinesq type. Admitting formally tanh D ∼ D for small frequencies and substituting D instead of tanh D to the nonlinear part of Equations (1.1), one comes to the system regarded in [15]. Hur and Pandey have proved the Benjamin-Feir instability [15] of periodic travelling waves for their system, which makes it valuable. If one in addition formally discards the term η∂ x u in the system given in [15], then a new alternative system turns out to be locally well-posed and features wave breaking [16]. However, the latter does not belong to the class of Boussinesq-Whitham models since nonlinear non-dispersive terms have been neglected.We would like to pay special attention to a system that was not considered in [10] but was introduced by Duchêne, Israwi and Talhouk [11]. They modified the bi-layer Green-Naghdi model improving the frequency dispersion. In fact, their system is also linearly fully dispersive, which makes it a close relative to System (1.1). Note that their system is Hamiltonian as well. Moreover, they have justified the Green-Naghdi modification proving well-posedness, consistency and convergence to the full water wave problem in the Boussinesq regime [11]. In addition, consistency of Hamiltonian structure is shown, so that energy levels of the approximate model can be compared with the full water energy. Existence of solitary waves for their system is also proved in [12]. Returning to the system regarded by Pei and Wang [22], we should notice that a question of existence of solitary waves for it, is closed as well [21]. Finally, we point out that well-posedness of the modified Green-Naghdi model is satisfactory, in the sense that it needs neither surface tension nor any non-physical initial condition. All this together makes it a promising system. And indeed, as noticed in [11], their modification gives more reliable results when it comes to large-frequency Kelvin-Helmholtz instabilities than other models of the Green-Naghdi type.On the contrary, System (1.1) has a couple of advantages compared with the modified Green-Naghdi model [11]. Firstly, it is derived, though not rigorously, from the Zakharov-Craig-Sulem formulation, and as a result one knows the relation between variables (η, v) and those describing the full potential fluid flow [10]. As to the modification discussed, it is presented in variables where the first one has the meaning of the surface...

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Equations (1.1) were firstly proposed and studied numerically in [10]. Later in [9] the first proof of local well-posedness based on an energy method and a compactness argument was given.…”

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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On well-posedness of a dispersive system of the Whitham–Boussinesq type

Dinvay

2019

Applied Mathematics Letters

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Comparison between Boussinesq‐ and Whitham–Boussinesq‐type systems

Deneke

Dufera

Tesfahun

2022

Math Methods in App Sciences

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In this paper, we consider a Whitham–Boussinesq‐type system modeling surface water waves of an inviscid incompressible fluid layer. The system describes the evolution with time of surface waves of a liquid layer in the two‐dimensional physical space. Using fixed point argument, we prove that the system is locally well‐posed on the time scale of order scriptO()1false/ϵ$$ \mathcal{O}\left(1/\sqrt{\epsilon}\right) $$, where ϵ>0$$ \epsilon &gt;0 $$ is the shallowness parameter measuring the ratio of amplitude of the wave to mean depth of fluid. We also show that the solution to the Whitham–Boussinesq system approximates the solution of a Boussinesq system on the time scale of order scriptO()1false/ϵ$$ \mathcal{O}\left(1/\sqrt{\epsilon}\right) $$.

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