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.

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.

scholarly journals A Hirota bilinear equation for Painlevé transcendents PIV, PII and PI

2018 ◽  
Vol 07 (04) ◽  
pp. 1840001
Author(s):  
A. N. W. Hone ◽  
F. Zullo
Keyword(s):  
Taylor Series ◽  
Taylor Expansion ◽  
Tau Function ◽  
Tau Functions ◽  
Sigma Function ◽  
Special Cases ◽  
Hirota Bilinear Equation ◽  
Order Four ◽  
Hirota Bilinear ◽  
Bilinear Equation

We present some observations on the tau-function for the fourth Painlevé equation. By considering a Hirota bilinear equation of order four for this tau-function, we describe the general form of the Taylor expansion around an arbitrary movable zero. The corresponding Taylor series for the tau-functions of the first and second Painlevé equations, as well as that for the Weierstrass sigma function, arise naturally as special cases, by setting certain parameters to zero.

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.

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.

scholarly journals Rational wave solutions to a generalized (2+1)-dimensional Hirota bilinear equation

2020 ◽  
Vol 15 ◽  
pp. 61 ◽  
Author(s):  
K. Hosseini ◽  
M. Mirzazadeh ◽  
M. Aligoli ◽  
M. Eslami ◽  
J.G. Liu
Keyword(s):  
Nonlinear Waves ◽  
Soliton Solutions ◽  
Wave Solutions ◽  
Multiple Soliton Solutions ◽  
Simplified Hirota’S Method ◽  
Complexiton Solutions ◽  
Hirota Bilinear Equation ◽  
Different Types ◽  
Hirota Bilinear ◽  
Bilinear Equation

A generalized form of (2+1)-dimensional Hirota bilinear (2D-HB) equation is considered herein in order to study nonlinear waves in fluids and oceans. The present goal is carried out through adopting the simplified Hirota’s method as well as ansatz approaches to retrieve a bunch of rational wave structures from multiple soliton solutions to breather, rational, and complexiton solutions. Some figures corresponding to a series of rational wave structures are provided, illustrating the dynamics of the obtained solutions. The results of the present paper help to reveal the existence of rational wave structures of different types for the 2D-HB equation.

scholarly journals Resonant behavior of multiple wave solutions to a Hirota bilinear equation

2016 ◽  
Vol 72 (5) ◽  
pp. 1225-1229 ◽  
Author(s):  
Li-Na Gao ◽  
Xue-Ying Zhao ◽  
Yao-Yao Zi ◽  
Jun Yu ◽  
Xing Lü
Keyword(s):  
Multiple Wave ◽  
Wave Solutions ◽  
Hirota Bilinear Equation ◽  
Resonant Behavior ◽  
Hirota Bilinear ◽  
Bilinear Equation
Sign in / Sign up

Export Citation Format

Share Document