Multi-lump formations from lump chains and plane solitons in the KP1 equation

Localized wave solutions and interactions of the (2+1)-dimensional Hirota–Satsuma–Ito equation

“…In the field of nonlinear wave, finding various of localized wave solutions of the NPDEs has been a focal point of research for many scholars. Examples of such solutions include soliton solutions, [7][8][9][10][11][12][13][14][15] breather wave solutions, [16][17][18][19][20] lump solutions, [21][22][23][24][25] rogue wave solutions, [26,27] and solitoncnoidal wave interaction solutions. [28,29] In the ocean, lump solutions are commonly used to describe phenomena such as breaking waves in the ocean and sharp peaks in tides.…”

Lump and soliton on certain spatially-varying backgrounds for an integrable (3+1) dimensional fifth-order nonlinear oceanic wave model

“…The Hirota bilinear method is directly linked with truncated Painlevé approach, where the later is successfully utilized to a variety of constant and variable coefficient soliton models to identify their integrability property and to obtain various nonlinear wave structures through auto-Bäcklund and hetero-Bäcklund transformations including Boussinesq-Burgers system [34], Whitham-Broer-Kaup-like system [35], variable coefficient generalized Burgers system [36] and the generalized variable-coefficient KdV-modified KdV equation [37]. Particularly, this Hirota method is successfully used to identify certain interesting dynamics of multi-lumps and lump chains in the framework of the KP1 model [38][39][40]. Interestingly, in addition to the above approaches, some recent methodologies such as probabilistic approaches [41,42] and deep learning techniques [43][44][45] attract much interest in understanding different nonlinear systems in different perspectives.…”

Some Exact Solutions to Generalized Kadomtsev-Petviashvili Equation

“…[7,11,12]. In recent years, the research on the lump solution of Equation (3) has been very hot, mainly focusing on the normal scattering of lump waves [13], anomalous scattering between lump waves [14][15][16][17][18], and bound states of lump waves [19]. The reports on anomalous scattering focus on the diversity of scattering patterns, such as triangular patterns, and polygonal patterns [17].…”