Transformed nonlinear waves, state transitions and modulation instability in a three-component AB model for the geophysi
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ORIGINAL PAPER
Transformed nonlinear waves, state transitions and modulation instability in a three-component AB model for the geophysical flows Han-Song Zhang · Lei Wang Xi-Yang Xie
· Xin Wang ·
Received: 12 May 2020 / Accepted: 15 September 2020 © Springer Nature B.V. 2020
Abstract In this paper, we investigate a threecomponent AB model, which characterizes the baroclinic instability processes in the geophysical flows. Via the Darboux transformation, the breather solutions are derived. Then, we study the state transition and find that the breather solutions can be transformed into different kinds of stationary nonlinear waves, including the anti-dark soliton, multi-peak soliton, M-shaped soliton, W-shaped solitons and periodic waves. Moreover, by virtue of the second-order transformed solution, various nonlinear wave complexes are presented. Finally, we unveil the relationship between the modulation instability and state transition and show the existence regions for the transformed waves. Keywords The three-component AB model · State transition · M-shaped soliton · W-shaped soliton · Multi-peak soliton · Periodic wave · Nonlinear wave complexes · Modulation instability
H. Zhang, L. Wang: These authors are co-first authors. H. Zhang · L. Wang (B) School of Mathematics and Physics, North China Electric Power University, Beijing 102206, China e-mail: [email protected] X. Wang College of Science, Zhongyuan University of Technology, Zhengzhou 450007, China X. Xie Department of Mathematics and Physics, North China Electric Power University, Baoding 071003, China
1 Introduction Nonlinear wave equations have been used to describe various nonlinear models in fluid dynamics [1,2], optics [3–6], plasma astrophysics [7], Bose–Einstein condensations (BECs) [8,9] and other fields [10–20]. Recently, two kinds of nonlinear waves have attracted broad attention, namely the rogue waves and breathers [21,22]. Rogue waves, which are initially to characterize the extreme wave events in deep oceans [1,2], have also been reported both in experimental observations [23] and theoretical predictions [24]. Those waves are thought to approach without casting a shadow and leave without leaving a trace [25–27]. Generally speaking, there are two kinds of breather structures, i.e., the Akhmediev breather [28] and Kuznetsov–Ma breather [29,30]. The former is localized in time and periodic in space, while the latter is localized in space and periodic in time [28–30]. Under certain limiting circumstance, both breathers can be considered as two potential prototypes of rogue waves in diverse physics fields [24,31– 33] and can be transformed into the Peregrine soliton [34]. Modulation instability (MI) is a characteristic of many dispersive nonlinear models, associated with the dynamical growth and evolution of regular perturbations on a nonzero background [35,36]. Recently, the close relationship between MI and rogue waves has been revealed. For example, the concept of baseband MI and passband MI were proposed by Baronio et
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