Trajectory equation of a lump before and after collision with line, lump, and breather waves for (2+1)-dimensional Kadomtsev–Petviashvili equation*

Abstract

Based on the hybrid solutions to (2+1)-dimensional Kadomtsev–Petviashvili (KP) equation, the motion trajectory of the solutions to KP equation is further studied. We obtain trajectory equation of a single lump before and after collision with line, lump, and breather waves by approximating solutions of KP equation along some parallel orbits at infinity. We derive the mathematical expression of the phase change before and after the collision of a lump wave. At the same time, we give some collision plots to reveal the obvious phase change. Our method proposed to find the trajectory equation of a lump wave can be applied to other (2+1)-dimensional integrable equations. The results expand the understanding of lump, breather, and hybrid solutions in soliton theory.

Resonance Y-shape solitons and mixed solutions for a (2+1)-dimensional generalized Caudrey–Dodd–Gibbon–Kotera–Sawada equation in fluid mechanics

“…Referring to an interesting work [38], we can know that the coordinates of lump's centre before and after interacting with resonant Y-shape soliton are given by parameters…”

Fission and Fusion Solutions to the (2+1)-Dimensional Generalized Konopelchenko–Dubrovsky–Kaup–Kupershmidt Equation Arising in Fluid Mechanics and Plasma Physics

“…And the trajectory of lump molecule is limited by {K 2m−1 , P 2m−1 , K 2m , p 2M }. The value of lump molecules before and after the collision with y-type molecules can be calculated according to the following formula [38,39]:…”

New various multisoliton kink‐type solutions of the (1 + 1)‐dimensional Mikhailov–Novikov–Wang equation

“…Over the past few years, many methods are being developed to expose new traveling solitary wave solutions for the nonlinear partial differential equation (PDE) representing the different areas of science and engineering. [1][2][3][4] Some of the analytical methods such as extended (G ′ /G)-expansion method, Darboux transformation, Pfaffian technique, sech-tanh method, sine-cosine method, Painlevé analysis, 5 Hirota bilinear method, [6][7][8][9][10][11][12][13][14][15] extended generalized Darboux transformation method, 16,17 Bäcklund transformation, and simplified Hirota's method [18][19][20][21][22][23][24] are used to solve different models involving nonlinear PDE. There is no specified method to solve all types nonlinear PDE.…”

“…Bilinear method is the method which is used to obtain analytical soliton solutions. [6][7][8][9][10][11][12][13][14][15] The Hirota bilinear method is applicable to those equations that take a bilinear form. The bilinear form of these nonlinear equation is highly nontrivial.…”

“…There is no specified method to solve all types nonlinear PDE. Bilinear method is the method which is used to obtain analytical soliton solutions 6–15 . The Hirota bilinear method is applicable to those equations that take a bilinear form.…”