The mixed solutions of the (2+1)-dimensional Hirota–Satsuma–Ito equation and the analysis of nonlinear transformed waves

In this paper, we obtain the N-soliton solution for the (2+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(2+1)$$\end{document}-dimensional Hirota–Satsuma–Ito equation by the Hirota bilinear method. On this basis, the breathers and lumps can be obtained using the complex conjugate parameter as well as the long wave limit method, and the mixed solutions containing them are investigated. Then, different nonlinear transformed waves are obtained from breathers and lumps under specific conditions, which include quasi-anti-dark soliton, M-shaped soliton, oscillation M-shaped soliton, multi-peak soliton, quasi-periodic soliton and W-shaped soliton. Finally, on the basis of the two-breather solutions, we discuss in detail the mixed solutions consisting of one breather and one nonlinear transformed wave, and the mixed solutions formed by two nonlinear transformed waves.