On the rotation-two-component Camassa–Holm system modelling the equatorial water waves

Fan, Lili; Gao, Hongjun; Liu, Yue

doi:10.1016/j.aim.2015.11.049

Cited by 51 publications

(34 citation statements)

References 29 publications

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“…The soliton solutions of the KB system are not yet completely studied, although there are some special soliton solutions obtained in [34,42,29]. Other 2-component integrable systems, that can match the model equations up to order δ 2 , are the 2-component Camassa-Holm system and the Zakharov-Ito system [13,28,27,24,23].…”

Section: Long Waves Approximationmentioning

confidence: 99%

The Dynamics of Flat Surface Internal Geophysical Waves with Currents

Compelli¹,

Ivanov²

2016

J. Math. Fluid Mech.

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A two-dimensional water wave system is examined consisting of two discrete incompressible fluid domains separated by a free common interface. In a geophysical context this is a model of an internal wave, formed at a pycnocline or thermocline in the ocean. The system is considered as being bounded at the bottom and top by a flatbed and wave-free surface respectively. A current profile with depth-dependent currents in each domain is considered. The Hamiltonian of the system is determined and expressed in terms of canonical wave-related variables. Limiting behaviour is examined and compared to that of other known models. The linearised equations as well as long-wave approximations are presented.Mathematics Subject Classification (2010): Primary 35Q35; Secondary 37K05, 74J30

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Section: Long Waves Approximationmentioning

confidence: 99%

The Dynamics of Flat Surface Internal Geophysical Waves with Currents

Compelli¹,

Ivanov²

2016

J. Math. Fluid Mech.

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“…Similar with system (2), the generalized two-component rotation b-family system (6) (or equivalent system (1)) is suitable for applying Kato's theory [30], and following the analogous proof to [18,19] (with some slight modifications), we thus have the well-posedness theorem.…”

Section: Proposition 1 ([3])mentioning

confidence: 98%

“…for t > 0 and x ∈ R. In the above system, u(x, t) is a horizontal velocity, ρ(t, x) is related to the free surface elevation from equilibrium, the parameter A describes a linear underlying shear flow, σ is a real dimensionless constant, providing the competition or balance, in fluid convection between nonlinear steepening and amplification due to stretching, µ is a non-dimensional parameter, and Ω is a real number characterizing the constant rotational speed of the Earth [19]. As we all 2476 MEILING YANG, YONGSHENG LI AND ZHIJUN QIAO known, the Earth's rotation profoundly affects the dynamics of the atmosphere and of the ocean.…”

mentioning

confidence: 99%

“…for t > 0 and x ∈ R. The R2CH system is derived by Fan-Gao-Liu [19] by using the approach in the spirit of Ivanov's asymptotic perturbation analysis for the governing equations of two-dimensional rotational gravity water waves [28]. There are at least three conservation laws for the system (2) E(u, ρ) =…”

mentioning

confidence: 99%

“…where the linear dispersion is absent. In [19], the authors provide an initial condition, which guarantees that the wave-breaking phenomena of (3) occur in a finite time. In [37], the authors also derive two finite-time blow-up results regarding the periodic case of (3).…”

mentioning

confidence: 99%

See 2 more Smart Citations

Persistence properties and wave-breaking criteria for a generalized two-component rotational b-family system

Yang¹,

Li²,

Zhang³

2020

Discrete &Amp; Continuous Dynamical Systems - A

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In this paper, we investigate a generalized two-component rotational b-family system arising in the rotating fluid with the effect of the Coriolis force. First, we study the persistence properties of the system in weighted L p-spaces, for a large class of moderate weights. Secondly, in order to overcome the difficulty arising from higher order nonlinearity and no conservation law, we take the advantage of the specially intrinsic structure of the system and make use of commutator estimate, and then derive two blow-up results for the strong solutions to the system.

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Nonuniform dependence of the R‐b‐family system in Besov spaces

2021

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In this paper, we consider a generalized rotation b‐family system (R‐b‐family system) which models the evolution of equatorial water waves. Based on local well‐posedness results and lifespan estimates, we establish sharpness of continuity on the data‐to‐solution map by showing that it is not uniformly continuous from Bp,rs×Bp,rs−1 to C(false[0,Tfalse];Bp,rs×Bp,rs−1). The proof of nonuniform dependence is based upon approximate solutions and Littlewood‐Paley decomposition theory.

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On the rotation-two-component Camassa–Holm system modelling the equatorial water waves

Cited by 51 publications

References 29 publications

The Dynamics of Flat Surface Internal Geophysical Waves with Currents

The Dynamics of Flat Surface Internal Geophysical Waves with Currents

Persistence properties and wave-breaking criteria for a generalized two-component rotational b-family system

Nonuniform dependence of the R‐b‐family system in Besov spaces

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