Resonance Y-type soliton and new hybrid solutions generated by velocity resonance for a (2+1)-dimensional generalized Bogoyavlensky–Konopelchenko equation in a fluid

Soliton molecules, Y-type soliton and complex multiple soliton solutions to the extended (3+1)-dimensional Jimbo-Miwa equation

“…To find the Y-type soliton solutions, we can adopt the following resonance conditions to the N-soliton solutions [54,[56][57][58][59][60]:…”

Novel soliton molecule solutions for the second extend (3+1)-dimensional Jimbo-Miwa equation in fluid mechanics

“…Based the bilinear form, like the lump molecule solution which does not exist in many (2+1)-dimensional integrable models, such as the (2+1)-dimensional generalized Bogoyavlensky-Konopelchenko model [38], Kadomtsev-Petviashvili (KP) system [39], (3+1)-dimensional nagative order KdV-CBS model [47], etc. But for KP systerm, lump molecules can be discovered by using the reduced version of the Grammian form [35,40].…”

“…The study of analytical solutions helps us to clarify the physical properties and behaviour of nonlinear equations, which are structural models for many physical phenomena [1]. There are a number of theoretical approaches to solving analytical solutions, such as the tanh-coth method [2][3][4], Riemann-Hilbert method [5][6][7], the inverse scattering method [8,9], Darboux transformation method [10][11][12][13], the Painlevé analysis [14], the generalized symmetry method [15], Hirota bilinear method [16,17,38] and others [18][19][20][21].…”

“…For equation (2), Wazwaz derived multiple soliton solutions [34], Guo et al presented four different localized waves and interaction solutions between lump solutions, line solitons, breathers and rogue waves using the Hirota bilinear method, Sun et al found lump and lump-kink solution [36], Xu et al constructed the resonance behavior with the aid of special parameter restrictions [37]. But as far as we know, no molecule solution of this equation has been studied, especially for the lump molecule solution, a special structural phenomenon that does not exist in most (2+1)-dimensional physical models by using long wave limit method [38,39]. All of findings in this paper can be used to explain some natural phenomena in the ocean waves and nonlinear optics.…”

Resonant Y-Type solutions, N-Lump waves, and hybrid solutions to a Ma-type model: a study of lump wave trajectories in superposition

“…By combining soliton molecules, fissionable waves, breather waves, and lump solutions, it is possible to obtain hybrid solutions in certain nonlinear wave equations. These hybrid solutions can exhibit complex behavior and provide a deeper understanding of the dynamics of the system [13][14][15] Researchers have been investigating elastic and inelastic collisions between lumps and other waves. However, despite this growing interest, there remains a lack of knowledge surrounding the trajectory of lumps before and after collisions.…”