Lump solutions with interaction phenomena in the (2+1)-dimensional Ito equation

2018 ◽  
Vol 32 (07) ◽  
pp. 1850104 ◽  
Author(s):  
Li Zou ◽  
Zong-Bing Yu ◽  
Shou-Fu Tian ◽  
Lian-Li Feng ◽  
Jin Li
Keyword(s):  
Necessary Conditions ◽  
Dynamic Properties ◽  
Quadratic Function ◽  
Graphical Analysis ◽  
Hirota’S Bilinear Method ◽  
Bilinear Method ◽  
Lump Solutions ◽  
Interaction Solutions ◽  
Sufficient And Necessary Conditions ◽  
Ito Equation

In this paper, we consider the (2+1)-dimensional Ito equation, which was introduced by Ito. By considering the Hirota’s bilinear method, and using the positive quadratic function, we obtain some lump solutions of the Ito equation. In order to ensure rational localization and analyticity of these lump solutions, some sufficient and necessary conditions are provided on the parameters that appeared in the solutions. Furthermore, the interaction solutions between lump solutions and the stripe solitons are discussed by combining positive quadratic function with exponential function. Finally, the dynamic properties of these solutions are shown via the way of graphical analysis by selecting appropriate values of the parameters.

Some Interaction Solutions of a Reduced Generalised (3+1)-Dimensional Shallow Water Wave Equation for Lump Solutions and a Pair of Resonance Solitons

2017 ◽  
Vol 72 (5) ◽  
pp. 419-424 ◽  
Author(s):  
Yao Wang ◽  
Mei-Dan Chen ◽  
Xian Li ◽  
Biao Li
Keyword(s):  
Shallow Water ◽  
Water Wave ◽  
Dynamic Properties ◽  
Quadratic Function ◽  
Translation Invariance ◽  
Bilinear Transformation ◽  
Lump Solutions ◽  
Interaction Solutions ◽  
Shallow Water Wave ◽  
Hirota Bilinear

AbstractThrough Hirota bilinear transformation and symbolic computation with Maple, a class of lump solutions, rationally localised in all directions in the space, to a reduced generalised (3+1)-dimensional shallow water wave (SWW) equation are prensented. The resulting lump solutions all contain six parameters, two of which are free due to the translation invariance of the SWW equation and the other four of which must satisfy a nonzero determinant condition guaranteeing analyticity and rational localisation of the solutions. Then we derived the interaction solutions for lump solutions and one stripe soliton and the result shows that the particular lump solutions with specific values of the involved parameters will be drowned or swallowed by the stripe soliton. Furthermore, we extend this method to a more general combination of positive quadratic function and hyperbolic functions. Especially, it is interesting that a rogue wave is found to be aroused by the interaction between lump solutions and a pair of resonance stripe solitons. By choosing the values of the parameters, the dynamic properties of lump solutions, interaction solutions for lump solutions and one stripe soliton and interaction solutions for lump solutions and a pair of resonance solitons, are shown by dynamic graphs.

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.

Localized interaction solution and its dynamics of the extended Hirota–Satsuma–Ito equation

2021 ◽  
pp. 2150313
Author(s):  
Jian-Ping Yu ◽  
Wen-Xiu Ma ◽  
Chaudry Masood Khalique ◽  
Yong-Li Sun
Keyword(s):  
Numerical Simulations ◽  
Rogue Wave ◽  
The Other ◽  
Hirota Bilinear Method ◽  
Bilinear Method ◽  
Interaction Solutions ◽  
Physical Phenomena ◽  
Wave Phenomena ◽  
Hirota Bilinear ◽  
Ito Equation

In this research, we will introduce and study the localized interaction solutions and th eir dynamics of the extended Hirota–Satsuma–Ito equation (HSIe), which plays a key role in studying certain complex physical phenomena. By using the Hirota bilinear method, the lump-type solutions will be firstly constructed, which are almost rationally localized in all spatial directions. Then, three kinds of localized interaction solutions will be obtained, respectively. In order to study the dynamic behaviors, numerical simulations are performed. Two interesting physical phenomena are found: one is the fission and fusion phenomena happening during the procedure of their collisions; the other is the rogue wave phenomena triggered by the interaction between a lump-type wave and a soliton wave.

Lumps, breathers, rogue waves and interaction solutions to a (3+1)-dimensional Kudryashov–Sinelshchikov equation

2020 ◽  
Vol 34 (12) ◽  
pp. 2050117 ◽  
Author(s):  
Xianglong Tang ◽  
Yong Chen
Keyword(s):  
Rogue Waves ◽  
Hirota Bilinear Method ◽  
Bilinear Method ◽  
Lump Solutions ◽  
Interaction Solutions ◽  
Long Wave ◽  
Wave Limit ◽  
Hirota Bilinear

Utilizing the Hirota bilinear method, the lump solutions, the interaction solutions with the lump and the stripe solitons, the breathers and the rogue waves for a (3[Formula: see text]+[Formula: see text]1)-dimensional Kudryashov–Sinelshchikov equation are constructed. Two types of interaction solutions between the lumps and the stripe solitons are exhibited. Some different breathers are given by choosing special parameters in the expressions of the solitons. Through a long wave limit of breathers, the lumps and rogue waves are derived.

Fission and fusion interaction phenomena of mixed lump kink solutions for a generalized (3+1)-dimensional B-type Kadomtsev–Petviashvili equation

2018 ◽  
Vol 32 (15) ◽  
pp. 1850161 ◽  
Author(s):  
Yaqing Liu ◽  
Xiaoyong Wen
Keyword(s):  
Shallow Water ◽  
Symbolic Computation ◽  
Water Wave ◽  
Hirota’S Bilinear Method ◽  
Bilinear Method ◽  
Lump Solutions ◽  
Shallow Water Wave ◽  
Petviashvili Equation ◽  
Kink Soliton ◽  
Interaction Phenomena

In this paper, a generalized (3[Formula: see text]+[Formula: see text]1)-dimensional B-type Kadomtsev–Petviashvili (gBKP) equation is investigated by using the Hirota’s bilinear method. With the aid of symbolic computation, some new lump, mixed lump kink and periodic lump solutions are derived. Based on the derived solutions, some novel interaction phenomena like the fission and fusion interactions between one lump soliton and one kink soliton, the fission and fusion interactions between one lump soliton and a pair of kink solitons and the interactions between two periodic lump solitons are discussed graphically. Results might be helpful for understanding the propagation of the shallow water wave.

scholarly journals Bäcklund Transformation and Exact Solutions to a Generalized (3 + 1)-Dimensional Nonlinear Evolution Equation

2022 ◽  
Vol 2022 ◽  
pp. 1-12
Author(s):  
Yali Shen ◽  
Ying Yang
Keyword(s):  
Evolution Equation ◽  
Nonlinear Evolution Equation ◽  
Nonlinear Evolution ◽  
Bäcklund Transformation ◽  
Dynamic Properties ◽  
Traveling Wave Solutions ◽  
Mixed Solution ◽  
Backlund Transformation ◽  
Bilinear Method ◽  
Interaction Solutions

In this article, a generalized (3 + 1)-dimensional nonlinear evolution equation (NLEE), which can be obtained by a multivariate polynomial, is investigated. Based on the Hirota bilinear method, the N-soliton solution and bilinear Bäcklund transformation (BBT) with explicit formulas are successfully constructed. By using BBT, two traveling wave solutions and a mixed solution of the generalized (3 + 1)-dimensional NLEE are obtained. Furthermore, the lump and the interaction solutions for the equation are constructed. Finally, the dynamic properties of the lump and the interaction solutions are described graphically.

Lump solutions and interaction behaviors to the (2+1)-dimensional extended shallow water wave equation

2018 ◽  
Vol 32 (31) ◽  
pp. 1850387 ◽  
Author(s):  
Wenguang Cheng ◽  
Tianzhou Xu
Keyword(s):  
Shallow Water ◽  
Water Wave ◽  
Dynamic Properties ◽  
Quadratic Function ◽  
Lump Solutions ◽  
Interaction Solution ◽  
Shallow Water Wave ◽  
Double Exponential ◽  
Combination Function ◽  
Bilinear Equation

In this paper, the exact solutions to the (2[Formula: see text]+[Formula: see text]1)-dimensional extended shallow water wave (SWW) equation are investigated by using its bilinear form and ansatz techniques. Following the method given by Ma [Phys. Lett. A 379 (2015) 1975–1978], two classes of lump solutions are constructed by searching for positive quadratic function solutions to the associated bilinear equation. Furthermore, two kinds of interaction solutions between a lump and solitary waves are presented by taking the solution of the associated bilinear equation as a linear combination function of a quadratic function and the double exponential function, one of which is the interaction solution between a lump and an exponentially decayed soliton, and the other one is the interaction solution between a lump and an exponentially decayed twin soliton. Finally, some figures are given to illustrate the dynamic properties of these obtained solutions.

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.

scholarly journals Lump Solutions and Interaction Solutions for the Dimensionally Reduced Nonlinear Evolution Equation

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-9 ◽  
Author(s):  
Baoyong Guo ◽  
Huanhe Dong ◽  
Yong Fang
Keyword(s):  
Evolution Equation ◽  
Nonlinear Evolution Equation ◽  
Nonlinear Evolution ◽  
Quadratic Functions ◽  
Bilinear Method ◽  
Lump Solutions ◽  
Dynamical Characteristics ◽  
Interaction Solutions ◽  
Physical Quantities ◽  
Hirota Bilinear

In this paper, by means of the Hirota bilinear method, a dimensionally reduced nonlinear evolution equation is investigated. Through its bilinear form, lump solutions are obtained. We construct interaction solutions between lump solutions and one soliton solution by choosing quadratic functions and exponential function. Interaction solutions with the combinations of exponential functions and sine function are also given. Meanwhile, the figures of these solutions are plotted. The dynamical characteristics and properties of obtained solutions are discussed, respectively. The results show that the corresponding physical quantities and properties of nonlinear waves are associated with the values of the parameters.

scholarly journals Abundant lump solutions and interaction solutions of a (3+1)-D Kadomtsev-Petviashvili equation

2019 ◽  
Vol 23 (4) ◽  
pp. 2437-2445 ◽  
Author(s):  
Xiaoqing Gao ◽  
Sudao Bilige ◽  
Jianqing Lü ◽  
Yuexing Bai ◽  
Runfa Zhang ◽  
...  
Keyword(s):  
Lump Solution ◽  
Solution Properties ◽  
Hirota Bilinear Method ◽  
Bilinear Method ◽  
Lump Solutions ◽  
Interaction Solutions ◽  
Petviashvili Equation ◽  
Contour Plots ◽  
Hirota Bilinear

In this paper, abundant lump solutions and two types of interaction solutions of the (3+1)-D Kadomtsev-Petviashvili equation are obtained by the Hirota bilinear method. Some contour plots with different determinant values are sequentially given to show that the corresponding lump solution tends to zero when the deter-minant approaches to zero. The interaction solutions with special parameters are plotted to elucidate the solution properties.

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