Viewing the Solar System via a variable-coefficient nonlinear dispersive-wave system
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Xin-Yi Gao · Yong-Jiang Guo · Wen-Rui Shan
Viewing the Solar System via a variable-coefficient nonlinear dispersive-wave system
Received: 2 March 2020 / Revised: 21 May 2020 © Springer-Verlag GmbH Austria, part of Springer Nature 2020
Abstract Oceans crossing the Solar System attract people’s attention: the Earth, Enceladus, and Titan. On a variable-coefficient nonlinear dispersive-wave system for the shallow oceanic environment, our symbolic computation yields two non-auto-Bäcklund transformations and auto-Bäcklund transformations with some solitons, with regard to the wave elevation and surface velocity of the water wave, which depend on the variable coefficients. This paper could be of some use for the future oceanic studies in the Solar System. Oceans crossing the Solar System have attracted people’s attention: For the Earth, shallow oceanic and coastal investigations have been performed1 [1–22]. For the Enceladus and Titan, Cassini spacecraft has revealed the presence of possible global oceans of liquid water beneath the icy crusts [23–29]. For the long gravity water waves in a shallow oceanic environment, Refs. [22,30–40] have presented a constant-coefficient nonlinear dispersive-wave system: ξt + [(1 + ξ ) u]x = − 41 u x x x , (1) u t + u u x + ξx = 0, where, as functions of the variables x and t, u(x, t) represents the surface velocity of the water wave along the x direction, ξ(x, t) is the wave elevation, and the subscripts denote the partial derivatives. Soliton interactions for System (1) have been reported in [22]. System (1) has been shown to be integrable [30,31], reducible from a Painlevé-nonintegrable model for the harbor and coastal design [32,33], characterized with several Hamiltonian structures [30,31], and equivalent to a member of the Ablowitz–Kaup–Newell–Segur system [32,34,35]. Multi-soliton solutions for System (1) have been seen by the Vandermonde determinant [32,34,35] and different Darboux transformations [36–39]. Other Darboux transformations have been constructed through the Lax pair of System (1) [40]. More relevant nonlinear-mechanics new contributions have been given, e.g., in Refs. [41–52]. In this paper, we will perform the Bäcklund-transformation work on a variable-coefficient extension of System (1): ξt + {[α1 (t) + α2 (t)ξ ] u}x = −α3 (t)u x x x ,
(2.1)
1 Examples are the investigations on the interaction of linear modulated waves and unsteady dispersive hydrodynamic states with the applications to the shallow water waves [17], water-wave scattering by multiple thin vertical barriers [16], water-wave scattering by three thin vertical barriers [18], internal tides over a shallow ridge with a high-resolution downscaling regional ocean model [19], shallow overturning circulation in the Indian Ocean [20], and propagation of long-crested water waves [21].
X.-Y. Gao (B)· Y.-J. Guo · W.-R. Shan State Key Laboratory of Information Photonics and Optical Communications, and School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China E-mail
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