scholarly journals On solutions of the combined KdV-nKdV equation

2019 ◽  
Vol 30 (2) ◽  
pp. 33
Author(s):  
Mohammed Allami ◽  
A. K. Mutashar ◽  
A. S. Rashid
Keyword(s):  
Vries Equation ◽  
Special Functions ◽  
Elliptic Functions ◽  
Rational Functions ◽  
Trigonometric Functions ◽  
Exponential Functions ◽  
Jacobian Elliptic Functions ◽  
Negative Order ◽  
De Vries ◽  
Wave Reduction

The aim of this work is to deal with a new integrable nonlinear equation of wave propagation, the combined of the Korteweg-de vries equation and the negative order Korteweg-de vries equation (combined KdV-nKdV) equation, which was more recently proposed by Wazwaz. Upon using wave reduction variable, it turns out that the reduced combined KdV-nKdV equation is alike the reduced (3+1)-dimensional Jimbo Miwa (JM) equation, the reduced (3+1)-dimensional Potential Yu-Toda-Sasa-Fukuyama (PYTSF) equation and the reduced (3 + 1)¬dimensional generalized shallow water (GSW) equation in the trav¬elling wave. In fact, the four transformed equations belong to the same class of ordinary differential equation. With the benefit of a well known general solutions for the reduced equation, we show that sub¬jects to some scaling and change of parameters, a variety of families of solutions are constructed for the combined KdV-nKdV equation which can be expressed in terms of rational functions, exponential functions and periodic solutions of trigonometric functions and hyperbolic func¬tions. In addition to that the equation admits solitary waves, and double periodic waves in terms of special functions such as Jacobian elliptic functions and Weierstrass elliptic functions.

Nonlocal Symmetries, Explicit Solutions, and Wave Structures for the Korteweg–de Vries Equation

2016 ◽  
Vol 71 (8) ◽  
pp. 735-740
Author(s):  
Zheng-Yi Ma ◽  
Jin-Xi Fei
Keyword(s):  
Vries Equation ◽  
Kdv Equation ◽  
Elliptic Functions ◽  
Group Method ◽  
Physical Interaction ◽  
Nonlocal Symmetries ◽  
Interaction Solutions ◽  
The Kdv Equation ◽  
De Vries ◽  
Exact Invariant

AbstractFrom the known Lax pair of the Korteweg–de Vries (KdV) equation, the Lie symmetry group method is successfully applied to find exact invariant solutions for the KdV equation with nonlocal symmetries by introducing two suitable auxiliary variables. Meanwhile, based on the prolonged system, the explicit analytic interaction solutions related to the hyperbolic and Jacobi elliptic functions are derived. Figures show the physical interaction between the cnoidal waves and a solitary wave.

Breather solutions of a negative order modified Korteweg-de Vries equation and its nonlinear stability

2019 ◽  
Vol 383 (15) ◽  
pp. 1689-1697 ◽  
Author(s):  
Jingqun Wang ◽  
Lixin Tian ◽  
Yingnan Zhang
Keyword(s):  
Nonlinear Stability ◽  
Vries Equation ◽  
Negative Order ◽  
De Vries

scholarly journals Nonlocal Symmetry and Bäcklund Transformation of a Negative-Order Korteweg–de Vries Equation

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-10 ◽  
Author(s):  
Jinxi Fei ◽  
Weiping Cao ◽  
Zhengyi Ma
Keyword(s):  
Vries Equation ◽  
Bäcklund Transformation ◽  
Backlund Transformation ◽  
Nonlocal Symmetry ◽  
Linear Superposition ◽  
Residual Symmetry ◽  
Negative Order ◽  
De Vries ◽  
Finite Transformation ◽  
New Variables

The residual symmetry of a negative-order Korteweg–de Vries (nKdV) equation is derived through its Lax pair. Such residual symmetry can be localized, and the original nKdV equation is extended into an enlarged system by introducing four new variables. By using Lie’s first theorem, we obtain the finite transformation for the localized residual symmetry. Furthermore, we localize the linear superposition of multiple residual symmetries and construct n-th Bäcklund transformation for this nKdV equation in the form of the determinants.

Specific solutions of the generalized Korteweg-de Vries equation with possible physical applications

Open Physics ◽  
2005 ◽  
Vol 3 (1) ◽  
Author(s):  
Paul Bracken
Keyword(s):  
Fluid Dynamics ◽  
Optical Waveguide ◽  
Vries Equation ◽  
Elliptic Functions ◽  
Localized Modes ◽  
Intrinsic Localized Modes ◽  
Waveguide Arrays ◽  
De Vries ◽  
Physical Impact

AbstractSolutions for a type of generalized Korteweg-de Vries equation which should have physical impact will be determined here. These types of solutions should have applications in the study of intrinsic localized modes optical waveguide arrays and fluid dynamics. It is shown that trigonometric and hyperbolic solutions can be obtained by matching powers and coefficients of the independent terms in the equation after the assumed solution has been substituted. As well, solutions to the equation in terms of more complicated Jacobe elliptic functions are determined.

A Discrete Negative Order Potential Korteweg–de Vries Equation

2016 ◽  
Vol 71 (12) ◽  
pp. 1151-1158 ◽  
Author(s):  
Song-lin Zhao ◽  
Ying-ying Sun
Keyword(s):  
Exact Solutions ◽  
Vries Equation ◽  
Matrix Approach ◽  
Cauchy Matrix ◽  
Negative Order ◽  
De Vries ◽  
Continuous Equation ◽  
Multisoliton Solutions ◽  
The Continuum

AbstractWe investigate a discrete negative order potential Korteweg–de Vries (npKdV) equation via the generalised Cauchy matrix approach. Solutions more than multisoliton solutions of this equation are derived by solving the determining equation set. We also show the semidiscrete equation and continuous equation together with their exact solutions by considering the continuum limits.

ON ANALYTIC AND PERIODIC SOLUTIONS FOR A CLASS OF NONLINEAR OSCILLATORS

1993 ◽  
Vol 03 (02) ◽  
pp. 451-454 ◽  
Author(s):  
JASON A.C. GALLAS
Keyword(s):  
Dynamical System ◽  
Equations Of Motion ◽  
Elliptic Functions ◽  
Nonlinear Oscillators ◽  
Analytic Solutions ◽  
One Dimension ◽  
Trigonometric Functions ◽  
Periodic Functions ◽  
Jacobian Elliptic Functions ◽  
Natural Extensions

We use the equations of motion for a particle moving in one dimension under the action of forces which vary cubically with displacement to discuss differential equations associated with double-periodic Jacobian elliptic functions. We show that analytic solutions for this dynamical system can be given in terms of Jacobian elliptic functions. These periodic functions are natural extensions of the well-known trigonometric functions.

scholarly journals Fractal complex transform technology for fractal Kkorteweg-de Vries equation within a local fractional derivative

2016 ◽  
Vol 20 (suppl. 3) ◽  
pp. 841-845
Author(s):  
Jinze Xu ◽  
Zeng-Shun Chen ◽  
Jian-Hong Wang ◽  
Ping Cui ◽  
Yunru Bai
Keyword(s):  
Fractional Derivative ◽  
Traveling Wave ◽  
Vries Equation ◽  
Special Functions ◽  
Traveling Wave Solutions ◽  
Cantor Sets ◽  
Local Fractional Derivative ◽  
Linear Partial Differential Equations ◽  
Wave Solutions ◽  
De Vries

In this paper, we present the fractal complex transform via a local fractional derivative. The traveling wave solutions for the fractal Korteweg-de Vries equations within local fractional derivative are obtained based on the special functions defined on Cantor sets. The technology is a powerful tool for solving the local fractional non-linear partial differential equations.

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