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

Yulei Cao, Yi Cheng, Boris A. Malomed, Jingsong He*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

22 Scopus citations


In this paper, the (Formula presented.) -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 (Formula presented.) -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.

Original languageEnglish
Pages (from-to)694-723
Number of pages30
JournalStudies in Applied Mathematics
Issue number2
StatePublished - Aug 2021


FundersFunder number
National Science Foundation12071451, 11671219, 12071304, 11871446
Israel Science Foundation1286/17


    • Hirota bilinear method
    • KP hierarchy reduction method
    • PT-symmetric nonlocal Maccari system
    • rogue wave
    • semirational solutions


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