Modulational instability and higher-order rogue waves with parameters modulation in a coupled integrable AB system via the generalized Darboux transformation

Abstract

We study higher-order rogue wave (RW) solutions of the coupled integrable dispersive AB system (also called Pedlosky system), which describes the evolution of wave-packets in a marginally stable or unstable baroclinic shear flow in geophysical fluids. We propose its continuous-wave (CW) solutions and existent conditions for their modulation instability to form the rogue waves. A new generalized N-fold Darboux transformation (DT) is proposed in terms of the Taylor series expansion for the spectral parameter in the Darboux matrix and its limit procedure and applied to the CW solutions to generate multi-rogue wave solutions of the coupled AB system, which satisfy the general compatibility condition. The dynamical behaviors of these higher-order rogue wave solutions demonstrate both strong and weak interactions by modulating parameters, in which some weak interactions can generate the abundant triangle, pentagon structures, etc. Particularly, the trajectories of motion of peaks and depressions of profiles of the first-order RWs are explicitly analyzed. The generalized DT method used in this paper can be extended to other nonlinear integrable systems. These results may be useful for understanding the corresponding rogue-wave phenomena in fluid mechanics and related fields.

High-order rogue waves of a long-wave–short-wave model of Newell type

“…Since the higher-order RW was excited experimentally in wave tanks [9,10], a hierarchy of higher-order analytic RW solutions which indicate the superposition of elementary RW has been found in nonlinear integrable systems [11][12][13][14][15][16][17][18][19][20]. Moreover, in contrast to the scalar system, recent studies have shown that multicomponent coupled systems may allow some novel patterns of RW such as dark and four-petaled types [21][22][23][24][25][26][27][28][29][30][31][32][33].…”

General High-order Rogue Waves of the (1+1)-Dimensional Yajima–Oikawa System

“…Another major development of importance is the study of rogue waves in multicomponent coupled systems, as a lot of complex physical systems usually contain interacting wave components with different modes and frequencies [44][45][46][47][48][49][50][51][52][53][54][55][56][57][58][59]. As stated in Ref.…”

W-shaped soliton complexes and rogue-wave pattern transitions for the AB system

“…Here we note that, dynamics of the Peregrine rogue waves and the first-order W-shaped soliton in the component A have been studied in Refs. [31][32][33][34], so here we omit presenting them and just concentrate on the higher-order W-shaped solitons in the rest of this paper.…”

“…Wen and Yan investigated higher-order rogue wave solutions of Eq. (1) by applying the generalized (1, N − 1)-fold DT [32]. Tian et al considered rogue wave solutions of the inhomogeneous extension of Eq.…”