Lump waves, solitary waves and interaction phenomena to the (2 + 1)-dimensional Konopelchenko–Dubrovsky equation

Solitons for the modified $(2 + 1)$-dimensional Konopelchenko–Dubrovsky equations

“…Moreover, some Jacobi elliptic function solutions will degenerate solitons and triangular function solutions when the module approaches to 0 or 1. It shows that most of the solutions provided in this work are different to those presented in [1,[16][17][18][19][20][21][22][23][24]].…”

“…Based on the Hirota bilinear form of Eq. (1), the lump waves, solitary waves, as well as interaction between lump waves and solitary waves, have been obtained [18]. Based on the Hirota direct method and linear superposition principle, Eq.…”

Multi-Peak and Bright-Dark Solitary Wave Solutions of Higher-Order Dispersive Nonlinear Schrödinger Equations and their Applications

“…With the development of nonlinear science, the accurate solutionsjof nonlinearjpartial differentialjequations (PDEs) have been widely concerned. It is well known that many of the natural sciences and engineering problems are attributed to the study of nonlinear PDEs, the nonlinear PDEs of study has important value in theory and research, because the exact solution can explain some physicaljphenomena in the naturaljsciences [1][2][3][4][5][6][7][8][9][10][11]. However, due to its nonlinear complexity, it is difficult to obtain useful analytic solutions for nonlinear PDEs, and it is difficult to find a suitable and universal method.…”

Dynamics of localized waves and interaction solutions for the $(3+1)$-dimensional B-type Kadomtsev–Petviashvili–Boussinesq equation

“…It is well known that some special type of exact solutions [1][2][3][4][5][6][7], including soliton (it has ionic and stability properties), lump (localized in all directions in the space), breather (localized in one certain direction with periodic structure), and rogue wave (localized in both time and space) of nonlinear evolution equations (NLEEs) depict many physical scenarios occurring in diverse areas of physics. In the past few decades, these exact solutions of NLEEs, such as the KP equation [8], the Konopelchenko-Dubrovsky equation [9], the potential Yu-Toda-Sasa-Fukuyama equation [10], and the (3 + 1)-dimensional Hirota bilinear equation, have been studied [11,12]. Meanwhile, several effective methods have been established by mathematicians and physicists to obtain the exact solutions of NLEEs, for instance, Painlevé analysis [13], Hirota bilinear method [14][15][16][17][18], Darboux transformation (DT) [19,20], and so on [21].…”