scholarly journals Lump Solutions and Resonance Stripe Solitons to the (2+1)-Dimensional Sawada-Kotera Equation

2017 ◽  
Vol 2017 ◽  
pp. 1-6 ◽  
Author(s):  
Xian Li ◽  
Yao Wang ◽  
Meidan Chen ◽  
Biao Li
Keyword(s):  
Symbolic Computation ◽  
Exponential Function ◽  
Bilinear Form ◽  
Quadratic Function ◽  
Hyperbolic Function ◽  
Lump Solutions ◽  
Interaction Solutions ◽  
Hirota Bilinear Form ◽  
Dynamical Properties ◽  
Hirota Bilinear

Based on the symbolic computation, a class of lump solutions to the (2+1)-dimensional Sawada-Kotera (2DSK) equation is obtained through making use of its Hirota bilinear form and one positive quadratic function. These solutions contain six parameters, four of which satisfy two determinant conditions to guarantee the analyticity and rational localization of the solutions, while the others are free. Then by adding an exponential function into the original positive quadratic function, the interaction solutions between lump solutions and one stripe soliton are derived. Furthermore, by extending this method to a general combination of positive quadratic function and hyperbolic function, the interaction solutions between lump solutions and a pair of resonance stripe solitons are provided. Some figures are given to demonstrate the dynamical properties of the lump solutions, interaction solutions between lump solutions, and stripe solitons by choosing some special parameters.

Interaction solutions for the (2+1)-dimensional Ito equation

2019 ◽  
Vol 33 (13) ◽  
pp. 1950167 ◽  
Author(s):  
Yaning Tang ◽  
Jinli Ma ◽  
Wenxian Xie ◽  
Lijun Zhang
Keyword(s):  
Periodic Function ◽  
Rational Function ◽  
Bilinear Form ◽  
Hyperbolic Function ◽  
Interaction Solution ◽  
Interaction Solutions ◽  
Hirota Bilinear Form ◽  
Dynamical Properties ◽  
Hirota Bilinear ◽  
Ito Equation

In this paper, two classes of interaction solutions of the (2[Formula: see text]+[Formula: see text]1)-dimensional Ito equation are studied in the case of Hirota bilinear form. As the results, the interaction solutions between the rational function and a periodic function as well as the interaction solution between the hyperbolic function and a periodic function are obtained. Based on the interaction solutions, a new transformation is proposed to analyze and discuss the influence of parameters. Furthermore, two kinds of lump solutions can be obtained via the limit behavior of the interaction solutions and the dynamical properties of these solutions are also illustrated.

Non-traveling lump solutions and mixed lump–kink solutions to (2+1)-dimensional variable-coefficient Caudrey–Dodd–Gibbon–Kotera–Sawada equation

2019 ◽  
Vol 33 (22) ◽  
pp. 1950262 ◽  
Author(s):  
Jing Wang ◽  
Hong-Li An ◽  
Biao Li
Keyword(s):  
Variable Coefficient ◽  
Quadratic Function ◽  
Lump Solution ◽  
Lump Solutions ◽  
Interaction Solutions ◽  
Hyperbolic Cosine ◽  
Hirota Bilinear Form ◽  
Dimensional Variable ◽  
Hyperbolic Cosine Function ◽  
Hirota Bilinear

Through Hirota bilinear form and symbolic computation with Maple, we investigate some non-traveling lump and mixed lump–kink solutions of the (2[Formula: see text]+[Formula: see text]1)-dimensional variable-coefficient Caudrey–Doddy–Gibbon–Kotera–Sawada equation by an extended method. Firstly, the non-traveling lump solutions are directly obtained by taking the function [Formula: see text] as a quadratic function. Secondly, we can get the interaction solutions for a lump solution and one kink solution by taking [Formula: see text] as a combination of quadratic function and exponential function. Finally, the interaction solutions between a lump solution and a pair of kinks solution can be derived by taking [Formula: see text] as a combination of quadratic function and hyperbolic cosine function. The dynamic phenomena of the above three types of exact solutions are demonstrated by some figures.

Dynamical analysis of solitary waves, lumps and interaction phenomena of a (2+1)-dimensional high-order nonlinear evolution equation

2019 ◽  
Vol 33 (16) ◽  
pp. 1950181 ◽  
Author(s):  
Bo Ren ◽  
Zhi-Mei Lou ◽  
Yong-Li Sun ◽  
Zhi-Wei He
Keyword(s):  
Solitary Waves ◽  
Bilinear Form ◽  
Nonlinear Evolution ◽  
Quadratic Function ◽  
High Order ◽  
Dynamical Analysis ◽  
Interaction Solutions ◽  
Variable Function ◽  
Hirota Bilinear Form ◽  
Hirota Bilinear

A (2[Formula: see text]+[Formula: see text]1)-dimensional high-order nonlinear evolution (HNE) equation is considered in this paper. A Hirota bilinear form of the HNE equation is constructed by the dependent variable function. Solitary waves are derived by solving the Hirota bilinear form of the HNE equation. Lump waves of the HNE equation are obtained by introducing a positive quadratic function. By mixing an exponential function or two exponential functions with a quadratic function, interaction solutions between a lump and a one-soliton, and between a lump and a two-soliton are presented. For the interaction solution between a lump and a two-soliton, this kind of solution can be considered as a special rogue wave. The propagation phenomena of these explicit solutions are illustrated by some graphs.

Abundant Lump Solution and Interaction Phenomenon of (3+1)-Dimensional Generalized Kadomtsev–Petviashvili Equation

2019 ◽  
Vol 20 (1) ◽  
pp. 33-40 ◽  
Author(s):  
Jianqing Lü ◽  
Sudao Bilige ◽  
Xiaoqing Gao
Keyword(s):  
Symbolic Computation ◽  
Bilinear Form ◽  
Lump Solution ◽  
Lump Solutions ◽  
Interaction Solution ◽  
Interaction Solutions ◽  
Petviashvili Equation ◽  
Contour Plots ◽  
Special Case ◽  
Interaction Phenomenon

AbstractIn this paper, with the help of symbolic computation system Mathematica, six kinds of lump solutions and two classes of interaction solutions are discussed to the (3+1)-dimensional generalized Kadomtsev–Petviashvili equation via using generalized bilinear form with a dependent variable transformation. Particularly, one special case are plotted as illustrative examples, and some contour plots with different determinant values are presented. Simultaneously, we studied the trajectory of the interaction solution.

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-type solutions for the (4+1)-dimensional Fokas equation via symbolic computations

2017 ◽  
Vol 31 (25) ◽  
pp. 1750224 ◽  
Author(s):  
Li Cheng ◽  
Yi Zhang
Keyword(s):  
Symbolic Computation ◽  
Bilinear Form ◽  
Symbolic Computations ◽  
Hirota Bilinear Form ◽  
Free Parameters ◽  
Almost All ◽  
Hirota Bilinear

Based on the Hirota bilinear form, two classes of lump-type solutions of the (4[Formula: see text]+[Formula: see text]1)-dimensional nonlinear Fokas equation, rationally localized in almost all directions in the space are obtained through a direct symbolic computation with Maple. The resulting lump-type solutions contain free parameters. To guarantee the analyticity and rational localization of the solutions, the involved parameters need to satisfy certain constraints. A few particular lump-type solutions with special choices of the involved parameters are given.

Interaction phenomenon to dimensionally reducedp-gBKP equation

2018 ◽  
Vol 32 (06) ◽  
pp. 1850074 ◽  
Author(s):  
Runfa Zhang ◽  
Sudao Bilige ◽  
Yuexing Bai ◽  
Jianqing Lü ◽  
Xiaoqing Gao
Keyword(s):  
Three Dimensional ◽  
Quadratic Function ◽  
Cosine Function ◽  
Interaction Solutions ◽  
Hyperbolic Cosine ◽  
Hirota Bilinear Form ◽  
Dynamical Features ◽  
Hyperbolic Cosine Function ◽  
Hirota Bilinear ◽  
Interaction Phenomenon

Based on searching the combining of quadratic function and exponential (or hyperbolic cosine) function from the Hirota bilinear form of the dimensionally reduced p-gBKP equation, eight class of interaction solutions are derived via symbolic computation with Mathematica. The submergence phenomenon, presented to illustrate the dynamical features concerning these obtained solutions, is observed by three-dimensional plots and density plots with particular choices of the involved parameters between the exponential (or hyperbolic cosine) function and the quadratic function. It is proved that the interference between the two solitary waves is inelastic.

scholarly journals Lump and Lump-Type Solutions of the Generalized (3+1)-Dimensional Variable-Coefficient B-Type Kadomtsev-Petviashvili Equation

2019 ◽  
Vol 2019 ◽  
pp. 1-5 ◽  
Author(s):  
Yanni Zhang ◽  
Jing Pang
Keyword(s):  
Symbolic Computation ◽  
Bilinear Form ◽  
Variable Coefficient ◽  
Three Dimensional ◽  
Petviashvili Equation ◽  
Hirota Bilinear Form ◽  
Contour Plots ◽  
Dimensional Variable ◽  
Hirota Bilinear

Based on the Hirota bilinear form of the generalized (3+1)-dimensional variable-coefficient B-type Kadomtsev-Petviashvili equation, the lump and lump-type solutions are generated through symbolic computation, whose analyticity can be easily achieved by taking special choices of the involved parameters. The property of solutions is investigated and exhibited vividly by three-dimensional plots and contour plots.

Painlevé Integrability and N-Soliton Solution for the Whitham- Broer-Kaup Shallow Water Model Using Symbolic Computation

2008 ◽  
Vol 63 (5-6) ◽  
pp. 253-260 ◽  
Author(s):  
Cheng Zhang ◽  
Bo Tian ◽  
Xiang-Hua Meng ◽  
Xing Lü ◽  
Ke-Jie Cai ◽  
...  
Keyword(s):  
Shallow Water ◽  
Symbolic Computation ◽  
Soliton Solution ◽  
Bilinear Form ◽  
Water Model ◽  
Shallow Water Model ◽  
Graphic Analysis ◽  
Hirota Bilinear Form ◽  
Hirota Bilinear ◽  
Singular Manifolds

With the help of symbolic computation, the Whitham-Broer-Kaup shallow water model is analyzed for its integrability through the Painlev´e analysis. Then, by truncating the Painlevé expansion at the constant level term with two singular manifolds, the Hirota bilinear form is obtained and the corresponding N-soliton solution with graphic analysis is also given. Furthermore, a bilinear auto-Bäcklund transformation is constructed for the Whitham-Broer-Kaup model, from which a one-soliton solution is presented.

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