Rigorous derivation from the water waves equations of some full dispersion shallow water models

Emerald, Louis

doi:10.48550/arxiv.2004.09240

Cited by 4 publications

(14 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…As such one expects an improved accuracy and range of validity as a model for surface waves compared to the long wave KdV approximation. More precisely, we anticipate that in the presence of weak nonlinearities, the Whitham equation stays close to the water waves equations, even if the dispersion effects are not small, as obtained in [10] for the bidirectional full dispersion systems.…”

Section: Motivationssupporting

confidence: 60%

“…The first method is based on an adaptation of the one used to derive the inviscid Burgers equations from the Nonlinear Shallow Water system using the Riemann invariants of the latter and requires, as in [15], that initial data are prepared to generate unidirectional waves. We show from it that the Whitham equation can be derived from the water waves equations at the order of precision µε, which is the same order as for the Whitham-Boussinesq equations when considering general initial data (see [10] for a rigorous derivation of the latter equations in the shallow water regime). From this derivation, we prove that solutions of the water waves equations, associated with well-prepared initial data, can be approximated in the shallow water regime, to within order µεt in a time scale of order ε −1 , by a right or left propagating wave solving a Whitham-type equation.…”

Section: Motivationsmentioning

confidence: 86%

“…for which we know the rigorous derivation from the water waves equations (see [10]). This is the main idea to prove this result.…”

Section: Herementioning

confidence: 99%

See 2 more Smart Citations

Rigorous derivation of the Whitham equations from the water waves equations in the shallow water regime

Emerald

2021

Preprint

Self Cite

View full text Add to dashboard Cite

We derive the Whitham equations from the water waves equations in the shallow water regime using two different methods, thus obtaining a direct and rigorous link between these two models. The first one is based on the construction of approximate Riemann invariants for a Whitham-Boussinesq system and is adapted to unidirectional waves. The second one is based on a generalisation of Birkhoff's normal form algorithm for almost smooth Hamiltonians and is adapted to bidirectional propagation. In both cases we clarify the improved accuracy on the fully dispersive Whitham model with respect to the long wave Korteweg-de Vries approximation.

show abstract

Section: Motivationssupporting

confidence: 60%

Section: Motivationsmentioning

confidence: 86%

See 1 more Smart Citation

Rigorous derivation of the Whitham equations from the water waves equations in the shallow water regime

Emerald

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…As a final remark let us point out that Theorem 1 can be applied to a more general Whitham-Boussinesq type system as one introduced in Remark 5.3 of [11]. It is possible to show that a linearization of System (1.1) around a solitary wave solution leads to an operator having both positive and negative essential spectrum.…”

Section: Proofmentioning

confidence: 95%

“…Different particular versions of System (1.1) appeared in [1,2,7,11,13]. For some of them existence of solitary wave solutions was proved in [4,5,8,15].…”

Section: Introductionmentioning

confidence: 99%

Travelling waves in the Boussinesq type systems

Dinvay¹

2020

Preprint

View full text Add to dashboard Cite

Considered herein are a number of variants of the Boussinesq type systems modeling surface water waves. Such equations were derived by different authors to describe the two-way propagation of long gravity waves. A question of existence of special solutions, the so called solitary waves, is of a particular interest. There are a number of studies relying on a variational approach and a concentration-compactness argument. These proofs are technically very demanding and may vary significantly from one system to another. Our approach is based on the implicit function theorem, which makes the treatment easier and more unified.

show abstract

Rigorous derivation of the Whitham equations from the water waves equations in the shallow water regime

Emerald

2021

Nonlinearity

View full text Add to dashboard Cite

We derive the Whitham equations from the water waves equations in the shallow water regime using two different methods, thus obtaining a direct and rigorous link between these two models. The first one is based on the construction of approximate Riemann invariants for a Whitham–Boussinesq system and is adapted to unidirectional waves. The second one is based on a generalisation of Birkhoff’s normal form algorithm for almost smooth Hamiltonians and is adapted to bidirectional propagation. In both cases we clarify the improved accuracy on the fully dispersive Whitham model with respect to the long wave Korteweg–de Vries approximation.

show abstract

Rigorous derivation from the water waves equations of some full dispersion shallow water models

Cited by 4 publications

References 0 publications

Rigorous derivation of the Whitham equations from the water waves equations in the shallow water regime

Rigorous derivation of the Whitham equations from the water waves equations in the shallow water regime

Travelling waves in the Boussinesq type systems

Rigorous derivation of the Whitham equations from the water waves equations in the shallow water regime

Contact Info

Product

Resources

About