Multiple rogue wave solutions and the linear superposition principle for a (3 + 1)‐dimensional Kadomtsev–Petviashvili–Boussinesq‐like equation arising in energy distributions

2021 ◽  
Author(s):  
Jalil Manafian
Keyword(s):  
Rogue Wave ◽  
Superposition Principle ◽  
Linear Superposition ◽  
Energy Distributions ◽  
Wave Solutions ◽  
Linear Superposition Principle ◽  
Rogue Wave Solutions

Nonlinear dynamics behavior of the (2+1)-dimensional Sawada–Kotera equation

2019 ◽  
Vol 33 (29) ◽  
pp. 1950355
Author(s):  
Hong-Yi Zhang ◽  
Yu-Feng Zhang
Keyword(s):  
Nonlinear Dynamics ◽  
Rogue Wave ◽  
Superposition Principle ◽  
Linear Superposition ◽  
Ansatz Method ◽  
3 Dimensional ◽  
Resonant Wave ◽  
Wave Solutions ◽  
Complexiton Solutions ◽  
Rogue Wave Solutions

In this paper, we mainly analyze the nonlinear dynamics behavior of the (2[Formula: see text]+[Formula: see text]1)-dimensional Sawada–Kotera (S–K) equation, which can be usually used to describe shallow water phenomena from natural science. First, the multiple resonant wave and complexiton solutions are constructed with the help of the linear superposition principle, under different domain fields, such as real and complex domain fields, respectively. Next, we apply a new ansatz method to obtain a class of rogue wave solutions (one-rogue wave and two-rogue wave solutions). Finally, the 3-dimensional and 2-dimensional density graphs are plotted for the yielded results in the above texts to better illustrate the dynamics processes to them.

The Multiple Exp-Function Method and the Linear Superposition Principle for Solving the (2+1)-Dimensional Calogero–Bogoyavlenskii–Schiff Equation

2015 ◽  
Vol 70 (9) ◽  
pp. 775-779 ◽  
Author(s):  
Elsayed M.E. Zayed ◽  
Abdul-Ghani Al-Nowehy
Keyword(s):  
Solitary Wave ◽  
Exact Solutions ◽  
Function Method ◽  
Superposition Principle ◽  
Solitary Wave Solutions ◽  
Linear Superposition ◽  
Wave Solutions ◽  
Linear Superposition Principle

AbstractIn this article, the multiple exp-function method and the linear superposition principle are employed for constructing the exact solutions and the solitary wave solutions for the (2+1)-dimensional Calogero–Bogoyavlenskii–Schiff (CBS) equation. With help of Maple and by using the multiple exp-method, we can get exact explicit one-wave, two-wave, and three-wave solutions, which include one-soliton-, two-soliton-, and three-soliton-type solutions. Furthermore, we apply the linear superposition principle to find n-wave solutions of the CBS equation. Two cases with specific values of the involved parameters are plotted for each two-wave and three-wave solutions.

Multi and breather wave soliton solutions and the linear superposition principle for generalized Hietarinta equation

2022 ◽  
Author(s):  
S. Şule Şener Kiliç
Keyword(s):  
Asymptotic Behavior ◽  
Superposition Principle ◽  
Soliton Solutions ◽  
Asymptotic Behavior Of Solutions ◽  
Hirota’S Bilinear Method ◽  
Bilinear Method ◽  
Linear Superposition ◽  
Wave Solutions ◽  
Linear Superposition Principle ◽  
Soliton Wave Solutions

In this paper, we study the generalized ([Formula: see text])-dimensional Hietarinta equation which is investigated by utilizing Hirota’s bilinear method. Also, the bilinear form is obtained, and the N-soliton solutions are constructed. In addition, multi-wave and breather wave solutions of the addressed equation with specific coefficients are presented. Finally, under certain conditions, the asymptotic behavior of solutions is analyzed in both methods. Moreover, we employ the linear superposition principle to determine [Formula: see text]-soliton wave solutions for the generalized ([Formula: see text])-dimensional Hietarinta equation.

Classifying bilinear differential equations by linear superposition principle

2016 ◽  
Vol 30 (28n29) ◽  
pp. 1640029 ◽  
Author(s):  
Lijun Zhang ◽  
Chaudry Masood Khalique ◽  
Wen-Xiu Ma
Keyword(s):  
Dispersion Relation ◽  
Superposition Principle ◽  
Kp Equation ◽  
Linear Superposition ◽  
Wave Solutions ◽  
Linear Superposition Principle ◽  
Step Algorithm ◽  
Hirota Bilinear ◽  
Bkp Equation ◽  
Bilinear Equations

In this paper, we investigate the linear superposition principle of exponential traveling waves to construct a sub-class of [Formula: see text]-wave solutions of Hirota bilinear equations. A necessary and sufficient condition for Hirota bilinear equations possessing this specific sub-class of [Formula: see text]-wave solutions is presented. We apply this result to find [Formula: see text]-wave solutions to the (2+1)-dimensional KP equation, a (3+1)-dimensional generalized Kadomtsev–Petviashvili (KP) equation, a (3+1)-dimensional generalized BKP equation and the (2+1)-dimensional BKP equation. The inverse question, i.e., constructing Hirota Bilinear equations possessing [Formula: see text]-wave solutions, is considered and a refined 3-step algorithm is proposed. As examples, we construct two very general kinds of Hirota bilinear equations of order 4 possessing [Formula: see text]-wave solutions among which one satisfies dispersion relation and another does not satisfy dispersion relation.

Multiple-Wave Solutions to Generalized Bilinear Equations in Terms of Hyperbolic and Trigonometric Solutions

2017 ◽  
Vol 18 (5) ◽  
pp. 395-401
Author(s):  
Ömer Ünsal ◽  
Wen-Xiu Ma ◽  
Yujuan Zhang
Keyword(s):  
Necessary Conditions ◽  
Superposition Principle ◽  
Trigonometric Function ◽  
Linear Superposition ◽  
Multiple Wave ◽  
Wave Solutions ◽  
Trigonometric Function Solutions ◽  
Sufficient And Necessary Conditions ◽  
Linear Superposition Principle ◽  
Bilinear Equations

AbstractThe linear superposition principle is applied to hyperbolic and trigonometric function solutions to generalized bilinear equations. We determine sufficient and necessary conditions for the existence of linear subspaces of hyperbolic and trigonometric function solutions to generalized bilinear equations. By using weights, three examples are given to show applicability of our theory.

The twin properties of rogue waves and homoclinic solutions for some nonlinear wave equations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Wei Tan ◽  
Zhao-Yang Yin
Keyword(s):  
Differential Equations ◽  
Wave Equations ◽  
Rogue Wave ◽  
Nonlinear Partial Differential Equations ◽  
Rogue Waves ◽  
Homoclinic Solutions ◽  
Nonlinear Wave Equations ◽  
Bilinear Method ◽  
Wave Solutions ◽  
Rogue Wave Solutions

Abstract The parameter limit method on the basis of Hirota’s bilinear method is proposed to construct the rogue wave solutions for nonlinear partial differential equations (NLPDEs). Some real and complex differential equations are used as concrete examples to illustrate the effectiveness and correctness of the described method. The rogue waves and homoclinic solutions of different structures are obtained and simulated by three-dimensional graphics, respectively. More importantly, we find that rogue wave solutions and homoclinic solutions appear in pairs. That is to say, for some NLPDEs, if there is a homoclinic solution, then there must be a rogue wave solution. The twin phenomenon of rogue wave solutions and homoclinic solutions of a class of NLPDEs is discussed.

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