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 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.

Resonant multi-soliton solutions to two fifth-order KdV equations via the simplified linear superposition principle

2019 ◽  
Vol 33 (25) ◽  
pp. 1950299 ◽  
Author(s):  
Chun-Ku Kuo
Keyword(s):  
Superposition Principle ◽  
Soliton Solutions ◽  
Kp Equation ◽  
Linear Superposition ◽  
Kdv Equations ◽  
Linear Superposition Principle ◽  
Free Parameters ◽  
Fifth Order ◽  
Physical Phenomena ◽  
Soliton Wave Solutions

In this paper, the simplified linear superposition principle is presented and employed to handle two versions of the fifth-order KdV equations, called the (2[Formula: see text]+[Formula: see text]1)-dimensional Caudrey–Dodd–Gibbon (CDG) equation and the (3[Formula: see text]+[Formula: see text]1)-dimensional generalized Kadomtsev–Petviashvili (KP) equation, respectively. Two general forms of resonant multi-soliton solutions are formally obtained. The paper proceeds step-by-step with increasing detail about the derivation process. Firstly, illustrate the algorithms of the linear superposition principle which paves the way for solving the wave related numbers. Then, demonstrate its application that exposes the proposed approach provides enough freedom to construct resonant multi-soliton wave solutions. Finally, some graphical representations of obtained solutions are portrayed by taking some definite values to free parameters, which describe various versions of inelastic interactions of resonant multi-soliton waves. The associated propagations may be related to large variety of real physical phenomena.

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