Different wave structures for the completely generalized Hirota–Satsuma–Ito equation

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

“…By substituting f 1 and f 2 into equation (19), we obtain an expression that can be solved to find the phase shift…”

“…Next, we substitute pairs of f 1 terms along with the corresponding term of f 2 into equation (19). For example, combining the first and second terms of f 1 with the first term of f 2 and inserting them into equation (19) gives an expression that can be solved for A 1,2 . Similarly, we can find the coupling constants A 1,3 and A 2,3 using the same approach.…”

“…Several studies have aimed to determine solutions for equation 2. For instance, in [19], the authors identified multiple lump solutions, interactions between a lump and a solitary wave, and collisions of a lump with a breather wave. Likewise, [20] investigated multiple soliton and lump waves, as well as interactions between a lump and a soliton wave.…”

Painlevé analysis and higher-order rogue waves of a generalized (3+1)-dimensional shallow water wave equation

“…Moreover, under the vanishing effect of the parameter Γ 2 = 0, the model (1) comes under the family of Hirota-Satsuma equations. Recently, another (2+1)D Hirota-Satsuma-Ito equation is also investigated and several interesting nonlinear waves are reported including multiple lumps, lump-solitary waves and lump-periodic waves in [44], which can be reduced from the present (3+1)D HSI model (1) for Γ 2 = 0, Γ 1 = 1 and considering spatial-temporal transformation z + t = T. Moreover for the vanishing Γ 1 parameter (1) reduces to different soliton equations including Calogero-Bogoyavlenskii-Schiff, KP, BKP and Jimbo-Miwa equations as mentioned above. From these reports, one can understand that the considered (3+1)D HSI equation (1) is more general with much physical importance.…”