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.

Oscillatory Solutions for Lattice Korteweg-de Vries-Type Equations

2018 ◽  
Vol 73 (2) ◽  
pp. 91-98
Author(s):  
Wei Feng ◽  
Song-Lin Zhao
Keyword(s):  
Vries Equation ◽  
Sylvester Equation ◽  
Matrix Approach ◽  
Cauchy Matrix ◽  
Oscillatory Solutions ◽  
Lattice Potential ◽  
De Vries

AbstractBy imposing some shift relations on r which satisfies the Sylvester equation KM+MK=rtc, oscillatory solutions are presented for some lattice Korteweg-de Vries-type equations, including the lattice potential Korteweg de-Vires equation, lattice potential modified Korteweg de-Vires equation, and lattice Schwarzian Korteweg-de Vries equation. This is done through the generalised Cauchy matrix approach.

Analytical study of exact solutions of the nonlinear Korteweg–de Vries equation with space–time fractional derivatives

2018 ◽  
Vol 32 (02) ◽  
pp. 1850012 ◽  
Author(s):  
Jiangen Liu ◽  
Yufeng Zhang
Keyword(s):  
Exact Solutions ◽  
Fractional Derivatives ◽  
Analytical Study ◽  
Vries Equation ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Space Time ◽  
Strong Basis ◽  
De Vries

This paper gives an analytical study of dynamic behavior of the exact solutions of nonlinear Korteweg–de Vries equation with space–time local fractional derivatives. By using the improved [Formula: see text]-expansion method, the explicit traveling wave solutions including periodic solutions, dark soliton solutions, soliton solutions and soliton-like solutions, are obtained for the first time. They can better help us further understand the physical phenomena and provide a strong basis. Meanwhile, some solutions are presented through 3D-graphs.

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.

Comments on the use of the Korteweg-de Vries equation in the study of anharmonic lattices

1974 ◽  
Vol 339 (1616) ◽  
pp. 119-126 ◽  
Keyword(s):  
Differential Equation ◽  
Lattice Dynamics ◽  
Continuum Limit ◽  
Vries Equation ◽  
Wave Dispersion ◽  
Order Partial Differential Equation ◽  
Lennard Jones ◽  
Long Wave ◽  
De Vries ◽  
The Continuum

The use of the Korteweg-de Vries equation as the continuum limit of the equations describing the anharmonic motion of atoms in a lattice is examined in the light of the periodic solutions recently constructed by Askar (1973). It is shown that the Korteweg-de Vries equation does not exactly represent the behaviour of the nonlinear lattice in the limit of long waves and that a sixth-order partial differential equation gives a more accurate description of the lattice dynamics. The relative accuracies of two long wave dispersion relations derived from the Korteweg-de Vries equation are discussed and numerical results are presented for Morse, Born-Meyer and Lennard-Jones potentials. Askar’s paper contains several misprints and errors and the corrected forms of his equations are presented in the appendix.

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