Lump solutions of the BKP equation

1990 ◽  
Vol 147 (8-9) ◽  
pp. 472-476 ◽  
Author(s):  
C.R Gilson ◽  
J.J.C Nimmo
Keyword(s):  
Lump Solutions ◽  
Bkp Equation

Lump solutions to the BKP equation by symbolic computation

2016 ◽  
Vol 30 (28n29) ◽  
pp. 1640028 ◽  
Author(s):  
Jing-Yun Yang ◽  
Wen-Xiu Ma
Keyword(s):  
Symbolic Computation ◽  
Kp Equation ◽  
Lump Solutions ◽  
Positivity Condition ◽  
Independent Variables ◽  
Computation Process ◽  
Starting Point ◽  
Hirota Bilinear Form ◽  
Hirota Bilinear ◽  
Bkp Equation

Lump solutions are rationally localized in all directions in the space. A general class of lump solutions to the (2+1)-dimensional B-Kadomtsev–Petviashvili (BKP) equation is presented through symbolic computation with Maple. The Hirota bilinear form of the equation is the starting point in the computation process. Like the KP equation, the resulting lump solutions contain six arbitrary parameters. Two of the parameters are due to the translation invariances of the BKP equation with the independent variables, and the other four need to satisfy a nonzero determinant condition and the positivity condition, which guarantee analyticity and rational localization of the solutions.

Lump, periodic lump and interaction lump stripe solutions to the (2+1)-dimensional B-type Kadomtsev–Petviashvili equation

2018 ◽  
Vol 32 (07) ◽  
pp. 1850106 ◽  
Author(s):  
Pinxia Wu ◽  
Yufeng Zhang ◽  
Iqbal Muhammad ◽  
Qiqi Yin
Keyword(s):  
Energy Distribution ◽  
Bilinear Form ◽  
Dynamic Process ◽  
Propagation Direction ◽  
Horizontal Velocity ◽  
Lump Solutions ◽  
Petviashvili Equation ◽  
Hirota’S Bilinear Form ◽  
Bkp Equation

In this paper, the Hirota’s bilinear form is employed to investigate the lump, periodic lump and interaction lump stripe solutions of the (2+1)-dimensional B-type Kadomtsev–Petviashvili (BKP) equation. Many results are obtained by dynamic process of figures. We analyze the propagation direction and horizontal velocity of lump solutions to find some constraint conditions which include positiveness and localization. In the process of the travel of the periodic lump solutions, it appears that the energy distribution is not symmetrical. The interaction lump stripe solutions of non-elastic indicate that the lump solitons are dropped and swallowed by the stripe soliton.

scholarly journals On the Exact Solutions of Two (3+1)-Dimensional Nonlinear Differential Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Fanning Meng ◽  
Yongyi Gu
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Shallow Water ◽  
Traveling Wave ◽  
Nonlinear Differential Equations ◽  
Complex Method ◽  
Meromorphic Solutions ◽  
Complex Differential Equations ◽  
Bkp Equation

In this article, exact solutions of two (3+1)-dimensional nonlinear differential equations are derived by using the complex method. We change the (3+1)-dimensional B-type Kadomtsev-Petviashvili (BKP) equation and generalized shallow water (gSW) equation into the complex differential equations by applying traveling wave transform and show that meromorphic solutions of these complex differential equations belong to class W, and then, we get exact solutions of these two (3+1)-dimensional equations.

New integrable (2+1)- and (3+1)-dimensional shallow water wave equations: multiple soliton solutions and lump solutions

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Abdul-Majid Wazwaz
Keyword(s):  
Shallow Water ◽  
Water Wave ◽  
Wave Equations ◽  
Soliton Solutions ◽  
Lump Solutions ◽  
Content Type ◽  
Shallow Water Wave ◽  
Multiple Soliton Solutions ◽  
Simplified Hirota’S Method ◽  
Shallow Water Wave Equations

Purpose This study aims to develop two integrable shallow water wave equations, of higher-dimensions, and with constant and time-dependent coefficients, respectively. The author derives multiple soliton solutions and a class of lump solutions which are rationally localized in all directions in space. Design/methodology/approach The author uses the simplified Hirota’s method and lump technique for determining multiple soliton solutions and lump solutions as well. The author shows that the developed (2+1)- and (3+1)-dimensional models are completely integrable in in the Painlené sense. Findings The paper reports new Painlevé-integrable extended equations which belong to the shallow water wave medium. Research limitations/implications The author addresses the integrability features of this model via using the Painlevé analysis. The author reports multiple soliton solutions for this equation by using the simplified Hirota’s method. Practical implications The obtained lump solutions include free parameters; some parameters are related to the translation invariance and the other parameters satisfy a non-zero determinant condition. Social implications The work presents useful algorithms for constructing new integrable equations and for the determination of lump solutions. Originality/value The paper presents an original work with newly developed integrable equations and shows useful findings of solitary waves and lump solutions.

Lump solutions to a generalized Kadomtsev–Petviashvili–Ito equation

2021 ◽  
pp. 2150437
Author(s):  
Liyuan Ding ◽  
Wen-Xiu Ma ◽  
Yehui Huang
Keyword(s):  
Symbolic Computation ◽  
Three Dimensional ◽  
Order Derivative ◽  
Hirota Bilinear Method ◽  
Bilinear Method ◽  
Lump Solutions ◽  
Dynamical Behaviors ◽  
Contour Plots ◽  
Hirota Bilinear ◽  
Ito Equation

A (2+1)-dimensional generalized Kadomtsev–Petviashvili–Ito equation is introduced. Upon adding some second-order derivative terms, its various lump solutions are explicitly constructed by utilizing the Hirota bilinear method and calculated through the symbolic computation system Maple. Furthermore, two specific lump solutions are obtained with particular choices of the parameters and their dynamical behaviors are analyzed through three-dimensional plots and contour plots.

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