Explicit solutions and numerical simulations for an asymptotic water waves model with surface tension

Haidar, Mohammad; Arwadi, Toufic El; Israwi, Samer

doi:10.1007/s12190-020-01333-8

Cited by 7 publications

(3 citation statements)

References 19 publications

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“…Another approach of dealing with nonlinear PDE's when looking for analytical exact solution is finding instead an explicit solution with correctors. Explicit solutions with correctors for asymptotic water waves models have been obtained in [17,20]. Actually, H s -consistent solutions are obtained to the models in the variable topography case using the analytic solution of the model in the flat topography configuration.…”

Section: Explicit Solution With Correctors Of Order O(ε 3 ) For the E...mentioning

confidence: 99%

Mathematical modeling and numerical analysis for the higher order Boussinesq system

et al. 2022

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This study deals with higher-ordered asymptotic equations for the water-waves problem. We considered the higher-order/extended Boussinesq equations over a flat bottom topography in the well-known long wave regime. Providing an existence and uniqueness of solution on a relevant time scale of order 1/ [[EQUATION]] and showing that the solution's behavior is close to the solution of the water waves equations with a better precision corresponding to initial data, the asymptotic model is well-posed in the sense of Hadamard. Then we compared several water waves solitary solutions with respect to the numerical solution of our model. At last, we solve explicitly this model and validate the results numerically.

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Section: Explicit Solution With Correctors Of Order O(ε 3 ) For the E...mentioning

confidence: 99%

Mathematical modeling and numerical analysis for the higher order Boussinesq system

et al. 2022

Self Cite

View full text Add to dashboard Cite

show abstract

“…Another approach of dealing with nonlinear PDE's when looking for analytical exact solution is finding instead an explicit solution with correctors. Explicit solutions with correctors for asymptotic water waves models have been obtained in [20,17]. Actually, H s -consistent solutions are obtained to the models in the variable topography case using the analytic solution of the model in the flat topography configuration.…”

Section: Explicit Solution With Correctors Of Order O(ε 3 ) For the E...mentioning

confidence: 99%

Mathematical modeling and numerical analysis for the higher order Boussinesq system

Bhorbatly¹,

Lteif²,

Israwi³

et al. 2021

Preprint

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This study deals with higher-ordered asymptotic equations for the water-waves problem. We considered the higher-order/extended Boussinesq equations over a flat bottom topography in the well-known long wave regime. Providing an existence and uniqueness of solution on a relevant time scale of order 1/ √ ε and showing that the solution's behavior is close to the solution of the water waves equations with a better precision corresponding to initial data, the asymptotic model is well-posed in the sense of Hadamard. Then we compared several water waves solitary solutions with respect to the numerical solution of our model. At last, we solve explicitly this model and validate the results numerically.

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“…Explicit solutions are obtained for Kdv and Camassa-Holm models with and without surface tension in [19] and [20]. In this work, we study the motion of an incompressible irrotational inviscid fluid, under the effect of both Coriolis force and surface tension, with a nonflat bottom.…”

Section: Introductionmentioning

confidence: 99%

Well‐posedness of the Green–Naghdi model for an uneven bottom in presence of the Coriolis effect and surface tension

Berjawi,

El Arwadi,

Israwi

et al. 2024

Stud Appl Math

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The objective of this work is to derive and analyze a Green–Naghdi model with Coriolis effect and surface tension in nonflat bottom geometry. Gui et al. derive a Green–Naghdi‐type model in flat bottom geometry under the gravity and Coriolis effect. Chen et al. proved the existence and uniqueness of solution in Sobolev space under a condition depending on the initial velocity and the Coriolis effect. In this paper, we provide a rigorous derivation of Green–Naghdi model under the influence of the two mentioned effects, with nonflat bottom. After that, the existence and construction of solutions for the derived model will be proved under two alternative conditions: the first one is the same condition as in Chen et al. and Berjawi et al. and the second one concerns only the Coriolis coefficient that supposed to be only of order . This existence and uniqueness result ameliorate the result of Chen et al. and Berjawi et al. in the sense that no condition on the velocity is needed. We also prove the continuity of the associated flow map.

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Explicit solutions and numerical simulations for an asymptotic water waves model with surface tension

Cited by 7 publications

References 19 publications

Mathematical modeling and numerical analysis for the higher order Boussinesq system

Mathematical modeling and numerical analysis for the higher order Boussinesq system

Mathematical modeling and numerical analysis for the higher order Boussinesq system

Well‐posedness of the Green–Naghdi model for an uneven bottom in presence of the Coriolis effect and surface tension

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