Rogue waves and lumps on the nonzero background in the ‐symmetric nonlocal Maccari system

Abstract

In this paper, the scriptPT‐symmetric version of the Maccari system is introduced, which can be regarded as a two‐dimensional generalization of the defocusing nonlocal nonlinear Schrödinger equation. Various exact solutions of the nonlocal Maccari system are obtained by means of the Hirota bilinear method, long‐wave limit, and Kadomtsev–Petviashvili (KP) hierarchy method. Bilinear forms of the nonlocal Maccari system are derived for the first time. Simultaneously, a new nonlocal Davey‐Stewartson–type equation is derived. Solutions for breathers and breathers on top of periodic line waves are obtained through the bilinear form of the nonlocal Maccari system. Hyperbolic line rogue wave (RW) solutions and semirational ones, composed of hyperbolic line RW and periodic line waves are also derived in the long‐wave limit. The semirational solutions exhibit a unique dynamical behavior. Additionally, general line soliton solutions on constant background are generated by restricting different tau‐functions of the KP hierarchy, combined with the Hirota bilinear method. These solutions exhibit elastic collisions, some of which have never been reported before in nonlocal systems. Additionally, the semirational solutions, namely, (i) fusion of line solitons and lumps into line solitons and (ii) fission of line solitons into lumps and line solitons, are put forward in terms of the KP hierarchy. These novel semirational solutions reduce to 2N‐lump solutions of the nonlocal Maccari system with appropriate parameters. Finally, different characteristics of exact solutions for the nonlocal Maccari system are summarized. These new results enrich the structure of waves in nonlocal nonlinear systems, and help to understand new physical phenomena.

Doubly localized two‐dimensional rogue waves generated by resonant collision in Maccari system

“…where f, g, 𝛿 0 are given in (20). (II) After the interaction (𝑥, 𝑦 → +∞) Soliton 1 (𝜉 + 𝜉 * + f ≈ 0): From the above asymptotic analysis, we can seen that the shape, velocity and amplitude of the two solitons remain unchanged after collision.…”

“…RWs in two-dimensional systems have often line RWs, which are only localized in time and have infinite length. [18][19][20][21][22][23][24] Furthermore, twodimensional RWs, such as oceanic RWs, should be doubly localized in most physical settings. But, there are few reports on doubly localized RWs in high-dimensional systems.…”

“…However, the theoretical research 10–14 and experimental observation 15–17 of most doubly localized RWs are available in one‐dimensional nonlinear systems. RWs in two‐dimensional systems have often line RWs, which are only localized in time and have infinite length 18–24 . Furthermore, two‐dimensional RWs, such as oceanic RWs, should be doubly localized in most physical settings.…”

“…As the general form of Maccari system () is very complicated, its exact solution is reported in the special case of $$s<0$$ 49 . Recently, the nonlocal extension and resonant collision solutions between breathers and rational waves of Maccari system () have also been discussed 20,21 . For the best of our knowledge, the generating mechanism of many nonlinear waves in Maccari system is still open, including two‐dimensional RWs.…”

Higher rogue and rogue-soliton interaction solutions of a (2 + 1) dimensional nonlinear model in fluid mechanics

“…Rogue waves with short periods, large amplitudes, and extremely destructive waves have been studied frequently in the latest research of oceanography and physics. Rogue waves are unexpected large water waves that appear and disappear suddenly, with extremely steep peaks, and even pose a great threat to large ships [1][2][3][4][5][6][7]. Interaction solutions can be used to explain nonlinear phenomena, especially in climate modeling and prediction, weather forecasting, fluid dynamics, geophysics, etc Many scholars did a lot of work on the interaction phenomena and obtained many meaningful interaction solutions [8][9][10][11][12][13][14][15].…”