Complexiton and resonant multiple wave solutions 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]].…”

“…From the properties of the Jacobi elliptic functions, we know that when r → 1, for these solutions listed from (19) to (40), some of them will become zero or constants, the others will degenerate soliton-like solutions, the degenerated solitons for Eq. (2) are expressed by…”

“…Based on the Hirota direct method and linear superposition principle, Eq. (1) was investigated to have the complexiton and resonant multiple wave solutions [19]. By means of the variable separation method and improved mapping approach, new types of variable-separation solutions (including solitary wave solutions, periodic wave solutions, and rational function solutions) for Eq.…”

“…Motivated by these works [1,2,[19][20][21][22], the presented paper will study the following modified Konopelchenko and Dubrovsky (mKD) equations:…”

Multi-complexiton solutions of the (2+1)-dimensional asymmetrical Nizhnik-Novikov-Veselov equation

“…[12] which means a combination of exponential waves and trigonometric waves and corresponds to complex eigenvalues of associated characteristic problems. Several methods for complexitons have been developed [13][14][15][16], among which the Hirota direct method was proved a promising one [17][18][19][20][21].…”