The Fission and Disintegration of Internal Solitary Waves Moving over Two-Dimensional Topography

Djordjević, Vladan; Redekopp, L. G.

doi:10.1175/1520-0485(1978)008<1016:tfadoi>2.0.co;2

Cited by 206 publications

(156 citation statements)

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“…Indeed, such a critical ratio appears straightforwardly in the rigid lid configuration: if δ < δ c ≡ √ γ, then the interface wave is of depression type, and if δ > δ c , then the interface wave is of elevation type. This critical ratio is well known in the literature, and has led to many extended models, where a cubic nonlinear term becomes the major source of nonlinearity for δ ∼ δ c (see for example [15,18,20,21,26]). It is interesting to see that even if the respective polarity of the interface slow mode waves in the free surface configuration, and the interface waves in the rigid lid configuration follow qualitatively the same behavior, the value of the critical ratio is notably different.…”

Section: Polarity It Is Obvious That For K = λMmentioning

confidence: 86%

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Boussinesq/Boussinesq systems for internal waves with a free surface, and the KdV approximation

Duchêne¹

2011

ESAIM: M2AN

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Abstract. We study here some asymptotic models for the propagation of internal and surface waves in a two-fluid system. We focus on the so-called long wave regime for one-dimensional waves, and consider the case of a flat bottom. Choi and R. Camassa, J. Fluid Mech. 313 (1996) 83-103]. We study the wellposedness of such systems, and the asymptotic convergence of their solutions towards solutions of the full Euler system. Then, we provide a rigorous justification of the so-called KdV approximation, stating that any bounded solution of the full Euler system can be decomposed into four propagating waves, each of them being well approximated by the solutions of uncoupled Korteweg-de Vries equations. Our method also applies for models with the rigid lid assumption, using the Boussinesq/Boussinesq models introduced in [J.L. Bona, D. Lannes and J.-C. Saut, J. Math. Pures Appl. 89 (2008) 538-566]. Our explicit and simultaneous decomposition allows to study in details the behavior of the flow depending on the depth and density ratios, for both the rigid lid and free surface configurations. In particular, we consider the influence of the rigid lid assumption on the evolution of the interface, and specify its domain of validity. Finally, solutions of the Boussinesq/Boussinesq systems and the KdV approximation are numerically computed, using a Crank-Nicholson scheme with a predictive step inspired from [C.

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Section: Polarity It Is Obvious That For K = λMmentioning

confidence: 86%

“…We recover the classical KdV equations in the rigid lid configuration (see for example [14,15,26,33]). Following Section 3.2, one would obtain in the same way a rigorous justification for the KdV approximation, under the rigid lid assumption.…”

Section: The Models Under the Rigid Lid Assumptionmentioning

confidence: 99%

Boussinesq/Boussinesq systems for internal waves with a free surface, and the KdV approximation

Duchêne¹

2011

ESAIM: M2AN

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“…В работе Пелиновского [3], а также в последовавшей работе [28] изучалась трансформация солитона на донном уступе в рамках уравнения Кортевега-де Вриза (КдВ). При этом было показ-но, что при умеренных амплитудах падающей и прошедшей волн трансформацию на крае уступа можно рассчитывать приближенно по формулам Лэмба (2).…”

Section: трансформация поверхностных волн на донном уступеunclassified

“…Трансформация уединен-ных волн подробно исследовалась в работе Пелиновского с соавторами [29]. Как и в более ранних работах [3,28], авторы использовали линейную теорию для расчета амплитуд трансформирован-ных над уступом волн, используя формулы Лэмба (2). Для подтверждения выводов теории были выполнены расчеты динамики одиночной волны (солитона) в рамках различных численных схем: на основе уравнения КдВ, обобщенного уравнения Буссинеска, а также в рамках полнонелинейной модели Навье-Стокса.…”

Section: T G X T H T G T G H H X T X Tunclassified

“…При этом, как следу-ет из формул Лэмба (2), максимальное усиление амплитуды прошедшей волны по отоношению к падающей не может быть больше двух вблизи края уступа. Затем, по мере эволюции трансформиро-ванного импульса, из него образуются вторичные солитоны, при этом, согласно теории уравнения КдВ [3,28], максимальная амплитуда вторичного солитона может превысить амплитуду трансфор-мированного на уступе импульса не более, чем в два раза.…”

Section: T G X T H T G T G H H X T X Tunclassified

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Transformation of surface and internal waves at the bottom step

Kurkin¹,

Semin²,

Stepanyants³

2016

J Marine Sci Res Dev

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Similarity Solutions of Some Two‐Space‐Dimensional Nonlinear Wave Evolution Equations

Redekopp

1980

Stud Appl Math

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Similarity reductions of the two-space-dimensional versions of the KortewegdeVries, modified Korteweg-deVries, Benjamin-Davis-Ono, and nonlinear Schrodinger equations are presented, and some solutions of the reduced equations are discussed. Exact dispersive solutions of the two-dimensional KortewegdeVries equation are obtained, and the similarity solution of this equation is shown to be reducible to the second Painleve transcendent.

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The Fission and Disintegration of Internal Solitary Waves Moving over Two-Dimensional Topography

Cited by 206 publications

References 0 publications

Boussinesq/Boussinesq systems for internal waves with a free surface, and the KdV approximation

Boussinesq/Boussinesq systems for internal waves with a free surface, and the KdV approximation

Transformation of surface and internal waves at the bottom step

Similarity Solutions of Some Two‐Space‐Dimensional Nonlinear Wave Evolution Equations

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