On wave-breaking phenomena for a new generalized two-component shallow water wave system
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On wave-breaking phenomena for a new generalized two-component shallow water wave system Xiaofang Dong1 Received: 28 July 2020 / Accepted: 16 October 2020 © Springer-Verlag GmbH Austria, part of Springer Nature 2020
Abstract In this paper, we mainly devote to investigate a generalized two-component Camassa– Holm system which can be derived from shallow water wave in equatorial ocean regions by Hu and Liu (Phys D 391:87–110, 2019). Depending on different real-valued interval of the competition or balance index σ , we establish the new wave-breaking criteria and extend the earlier blow-up results for the system. Keywords Two-component system · Shallow water wave · Wave-breaking phenomena Mathematics Subject Classification 35A01 · 35B44 · 35B65 · 35G25
1 Introduction It is well known that there exists a two-layer fluid which contains a shallow nearsurface layer of relatively warm water and a deep layer of colder and denser water in the equatorial region [21]. The upper shallow water is less than 300 m deep [36] and usually the wavelength of surface waves can be 100 km or more. For example, the Rossby wave is a kind of the geophysical equatorial water waves which can be looked as a shallow-water wave whose wavelength can be 500 km. On the other hand, the Coriolis force generated by the rotation of the earth can not be ignored for some motions on an enormous range of spatial and temporal scales, such as the massive and persistent ocean current system. In the equatorial region, although the Coriolis force is small due to the smallness of the variation of latitude, the Coriolis
Communicated by Adrian Constantin.
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Xiaofang Dong [email protected] School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, People’s Republic of China
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X. Dong
force still affects the motion of waves [15,16]. Since the propagation of the equatorial flow is typically in one direction and the flow pattern is two-dimensional in a reasonable extent, the study of wave dynamics in the equatorial region of the Pacific Ocean has attracted the attention of many authors in the recent years, as confirmed through studies concerning mathematical modeling [10,15,16]. Some significant modifications have been made to explain the geophysical effects of the generalized Dullin–Gottwald– Holm system and the two-component Camassa–Holm system [14,22,30,35]. A generalized rotation two-component Camassa–Holm (CH) system in [29] which was derived as a model of shallow water wave propagating mainly in the equatorial ocean regions with the Coriolis effect caused by the Earth’s rotation can be expressed as ⎧ 1 2 2 ⎪ ⎨ m t + σ (2mu x + um x ) + 3(1 − σ )uu x + 2 (ρ )x + 2(ρ u)x −8(ρu)x + 4u x + 24β1 (u 2 ρ(ρ − 1))x + 4β2 (u 3 )x = 0, ⎪ ⎩ ρt + (ρu)x + 2uu x ρ + 8β1 (u 3 )x ρ = 0,
t > 0, x ∈ R, t > 0, x ∈ R, (1.1)
√ 2 2c4 −2c2 +8 where β1 = 5c48c−11 , c = 2 + 1− and is the constant Coriolis 2 , β2 = 8c3 frequency due to the Earth’s rotation. Here u(t, x) is the average fluid velocity in the x-direction and m = u − u x x is the
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