scholarly journals Two Different Classes of Wronskian Conditions to a (3 + 1)-Dimensional Generalized Shallow Water Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-10
Author(s):  
Yaning Tang ◽  
Pengpeng Su
Keyword(s):  
Shallow Water ◽  
Shallow Water Equation ◽  
Sufficient Conditions ◽  
Wronskian Technique ◽  
Rational Solutions ◽  
Bilinear Method ◽  
Wronskian Determinant ◽  
Linear Partial Differential Equations ◽  
Hirota Bilinear Equation ◽  
Hirota Bilinear

Based on the Hirota bilinear method and Wronskian technique, two different classes of sufficient conditions consisting of linear partial differential equations system are presented, which guarantee that the Wronskian determinant is a solution to the corresponding Hirota bilinear equation of a (3+1)-dimensional generalized shallow water equation. Our results show that the nonlinear equation possesses rich and diverse exact solutions such as rational solutions, solitons, negatons, and positons.

A (2+1)-dimensional shallow water equation and its explicit lump solutions

2019 ◽  
Vol 33 (07) ◽  
pp. 1950038 ◽  
Author(s):  
Solomon Manukure ◽  
Yuan Zhou
Keyword(s):  
Shallow Water Equation ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Bilinear Method ◽  
Lump Solutions ◽  
Contour Plots ◽  
Dimensional Equation ◽  
New Equation ◽  
Necessary And Sufficient ◽  
Hirota Bilinear

We introduce a new (2+1)-dimensional equation by modifying the potential form of the Calogero–Bogoyavlenskii–Schiff (CBS) equation. By applying the Hirota bilinear method, we construct explicit lump solutions to this new equation and establish necessary and sufficient conditions that guarantee that the solutions are analytic and rationally localized in all directions in space. We also depict the evolution of the profiles of some selected lump solutions with three-dimensional and contour plots. It is immediately observed that the lump solutions generated are solitary wave type solutions as is the case with the KP equation.

THE GENERALIZED WRONSKIAN SOLUTIONS OF THE INTEGRABLE VARIABLE-COEFFICIENT KORTEWEG-DE VRIES EQUATION

2011 ◽  
Vol 25 (32) ◽  
pp. 4615-4626 ◽  
Author(s):  
YI ZHANG ◽  
HAI-QIONG ZHAO ◽  
LING-YA YE ◽  
YI-NENG LV
Keyword(s):  
Fluid Dynamics ◽  
Exact Solutions ◽  
Variable Coefficient ◽  
Vries Equation ◽  
Sufficient Conditions ◽  
Rational Solutions ◽  
Wronskian Determinant ◽  
Linear Partial Differential Equations ◽  
De Vries ◽  
Bose Einstein

A broad set of sufficient conditions consisting of systems of linear partial differential equations are presented which guarantee that the Wronskian determinant is the solutions of the integrable variable-coefficient Korteweg-de Vries model from Bose–Einstein condensates and fluid dynamics. The generalized Wronskian solutions provide us with a comprehensive approach to construct many exact solutions including rational solutions, solitons, negatons, positons, and complexitons.

Rogue Waves and Hybrid Solutions of the Boussinesq Equation

2017 ◽  
Vol 72 (4) ◽  
pp. 307-314 ◽  
Author(s):  
Ji-Guang Rao ◽  
Yao-Bin Liu ◽  
Chao Qian ◽  
Jing-Song He
Keyword(s):  
Boussinesq Equation ◽  
Rogue Wave ◽  
Three Dimensional ◽  
Rogue Waves ◽  
Rational Solutions ◽  
Bilinear Method ◽  
Long Wave ◽  
Hybrid Solutions ◽  
Wave Limit ◽  
Hirota Bilinear

AbstractThe rational and semirational solutions in the Boussinesq equation are obtained by the Hirota bilinear method and long wave limit. It is shown that the rational solutions contain dark and bright rogue waves, and their typical dynamics are analysed and illustrated. The semirational solutions possess a range of hybrid solutions, and the hybrid of rogue wave and solitons are demonstrated in detail by the three-dimensional figures. Under certain parameter conditions, a new kind of semirational solutions consisted of rogue waves, breathers and solitons is discovered, which describes the dynamics of the rogue waves interacting with the breathers and solitons at the same time.

scholarly journals New Exact Solutions for the (3+1)-Dimensional Generalized BKP Equation

2016 ◽  
Vol 2016 ◽  
pp. 1-9 ◽  
Author(s):  
Jun Su ◽  
Genjiu Xu
Keyword(s):  
Exact Solutions ◽  
Evolution Equations ◽  
Soliton Solutions ◽  
Nonlinear Evolution Equations ◽  
Wronskian Technique ◽  
Rational Solutions ◽  
Wronskian Determinant ◽  
New Exact Solutions ◽  
Positon Solutions ◽  
Bkp Equation

The Wronskian technique is used to investigate a (3+1)-dimensional generalized BKP equation. Based on Hirota’s bilinear form, new exact solutions including rational solutions, soliton solutions, positon solutions, negaton solutions, and their interaction solutions are formally derived. Moreover we analyze the strangely mechanical behavior of the Wronskian determinant solutions. The study of these solutions will enrich the variety of the dynamics of the nonlinear evolution equations.

scholarly journals Abundant exact analytical solutions and novel interaction phenomena of the generalized (3+1)-dimensional shallow water equation

2021 ◽  
pp. 103-103
Author(s):  
Xiaomin Wang ◽  
Sudao Bilig ◽  
Yueyang Feng
Keyword(s):  
Shallow Water ◽  
Analytical Solutions ◽  
Shallow Water Equation ◽  
Solution Process ◽  
Soliton Solutions ◽  
High Order ◽  
Bilinear Method ◽  
Exact Analytical Solutions ◽  
Wave Solutions ◽  
Interaction Phenomena

This paper reveals abundant exact analytical solutions to the generalized (3+1)-dimensional shallow water equation. The generalized bilinear method is used in the solution process and the obtained solutions include the high-order lump-type solutions, the three-wave solutions, the breather solutions. The interaction between the high-order lump-type solutions and the soliton solutions is also elucidated. These solutions have greatly enriched the generalized (3+1)-dimensional shallow water equation in open literature.

Two-dimensional new coherent structures of lump-soliton solutions for the Mel’nikov equation

2021 ◽  
pp. 2150416
Author(s):  
Wei Liu ◽  
Yuan Meng ◽  
Xiaoyan Qiao
Keyword(s):  
Coherent Structures ◽  
Dimensional Space ◽  
Soliton Solutions ◽  
Two Dimensional ◽  
Rational Solutions ◽  
Bilinear Method ◽  
Dark Line ◽  
New Novel ◽  
Short Period ◽  
Hirota Bilinear

Under investigation in this paper are new novel coherent structures of two-dimensional lump-soliton for the Mel’nikov equation. The Hirota bilinear method and Kadomtsev–Petviashvili hierarchy reduction method are applied to construct a particular family of determinant semi-rational solutions exhibiting various coherent waves to the Mel’nikov equation. We first investigate some novel coherent waves, [Formula: see text]th-order lumps first appear from the [Formula: see text] dark line solitons and finally disappear into those [Formula: see text] dark line solitons after living on the constant background for a very short period. In contrast to the usual lump, those lumps in the coherent structures of lump-soliton are not only localized in two-dimensional space and but also localized in time.

Rational solutions and lump solutions to the (3+1)-dimensional Mel’nikov equation

2019 ◽  
Vol 34 (03) ◽  
pp. 2050033 ◽  
Author(s):  
Xuelin Yong ◽  
Xiaoyu Li ◽  
Yehui Huang ◽  
Wen-Xiu Ma ◽  
Yong Liu
Keyword(s):  
Three Dimensional ◽  
Explicit Representation ◽  
Hirota Bilinear Method ◽  
Matrix Elements ◽  
Rational Solutions ◽  
Bilinear Method ◽  
Lump Solutions ◽  
First Order ◽  
Special Value ◽  
Hirota Bilinear

In this paper, explicit representation of general rational solutions for the (3[Formula: see text]+[Formula: see text]1)-dimensional Mel’nikov equation is derived by employing the Hirota bilinear method. It is obtained in terms of determinants whose matrix elements satisfy some differential and difference relations. By selecting special value of the parameters involved, the first-order and second-order lump solutions are given and their dynamic characteristics are illustrated by two- and three-dimensional figures.

scholarly journals Hybrid Solutions of (3 + 1)-Dimensional Jimbo-Miwa Equation

2017 ◽  
Vol 2017 ◽  
pp. 1-15 ◽  
Author(s):  
Yong Zhang ◽  
Shili Sun ◽  
Huanhe Dong
Keyword(s):  
Three Dimensional ◽  
Rogue Waves ◽  
Lump Solution ◽  
Rational Solutions ◽  
Bilinear Method ◽  
Long Wave ◽  
Hybrid Solution ◽  
Hybrid Solutions ◽  
Wave Limit ◽  
Hirota Bilinear

The rational solutions, semirational solutions, and their interactions to the (3+1)-dimensional Jimbo-Miwa equation are obtained by the Hirota bilinear method and long wave limit. The hybrid solutions contain rogue wave, lump solution, and the breather solution, in which the breathers which are manifested as growing and decaying periodic line waves show different dynamics in different planes. Rogue waves are localized in time and are obtained theoretically as a long wave limit of breathers with indefinitely larger periods; they arise from a constant background at t≪0 and then disappear in the constant background when time goes on. More importantly, the interactions between some hybrid solutions are demonstrated in detail by the three-dimensional figures, such as hybrid solution between the stripe soliton and breather and hybrid solution between stripe soliton and lump solution.

Study of lump and lump-kink solitons of a coupled reduced Hirota bilinear equation

2020 ◽  
Vol 34 (22) ◽  
pp. 2050224
Author(s):  
Shun Wang ◽  
Chuanzhong Li ◽  
Zhenli Wang
Keyword(s):  
Sufficient Conditions ◽  
Quadratic Function ◽  
Quadratic Functions ◽  
Bulk Solution ◽  
Lump Solutions ◽  
Weakly Coupled ◽  
Hirota Bilinear Equation ◽  
Necessary And Sufficient ◽  
Hirota Bilinear ◽  
Bilinear Equation

By symbolic computation and searching for the solutions of the positive quadratic functions of the related bilinear equations, two kinds of lump solutions of the (3[Formula: see text]+[Formula: see text]1)-dimensional weakly coupled Hirota bilinear equation are derived, and the practicability of this method is verified. Then we add an exponential function to the original positive quadratic function, and obtain a new solution of the Hirota bilinear equation. The interaction between the lump solutions and lump-kink solutions is included in the new solution. On this basis, we give the possibility of adding multiple exponential functions. Finally, we give the coupled reduced Hirota bilinear equation lump-kink solitons by combining the above two methods. In order to ensure the analyticity and reasonable localization of the block, two sets of necessary and sufficient conditions are given for the parameters involved in the solution. The local characteristics and energy distribution of bulk solution are analyzed and explained.

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