Breather-to-soliton conversions and their mechanisms of the (2+1)-dimensional generalized Hirota–Satsuma–Ito equation

Different nonlinear wave structural solutions of the (2+1)-dimensional generalized Hirota–Satsuma–Ito equation are derived for the first time via adding transition condition, and the dynamics of transformed nonlinear waves are discussed. Firstly, breath waves composed of a solitary part and a periodic part are found by changing wave number values of the N-soliton solution into complex forms. These two parts are respectively described by a characteristic line. Interestingly, when the characteristic lines of two parts are parallel, various transformed nonlinear wave structures are explored. Then, the gradient relations, superposition theorem, local and oscillating properties, and the time-varying dynamics of the transformed solutions of this equation are discussed by analyzing characteristic lines and their displacements. Furthermore, we investigated the interactions between transformed solutions, and reveal that the shape change after the collision is the result of the change in the distance between the characteristic lines. Finally, through the translation and scaling of the characteristic lines, we propose a new translation transformation and a new scaling transformation applicable to above solutions. These results enable us to have a deeper understanding of the structure of the transformed solutions. Moreover, it is helpful for finding various types of solutions with position and range requirements.