Symbolic computation of conservation laws and exact solutions of a coupled variable-coefficient modified Korteweg–de Vries system

2016 ◽  
Vol 56 (4) ◽  
pp. 650-660 ◽  
Author(s):  
Abdullahi Rashid Adem ◽  
Chaudry Masood Khalique
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
Symbolic Computation ◽  
Variable Coefficient ◽  
De Vries

ON THE EXISTENCE OF INFINITE CONSERVATION LAWS OF A VARIABLE-COEFFICIENT KORTEWEG–DE VRIES MODEL WITH SYMBOLIC COMPUTATION

2011 ◽  
Vol 25 (20) ◽  
pp. 1683-1689
Author(s):  
HONG-WU ZHU ◽  
BO TIAN ◽  
CHUN-YI ZHANG
Keyword(s):  
Conservation Laws ◽  
Symbolic Computation ◽  
Gravity Waves ◽  
Variable Coefficient ◽  
Miura Transformation ◽  
Scattering Transform ◽  
De Vries ◽  
Bose Einstein ◽  
Invariant Transformation ◽  
Dissipative Terms

Under investigation in this paper is a generalized variable-coefficient Korteweg–de Vries (vcKdV) model with external-force and perturbed/dissipative terms, which can describe various real dynamical processes of physics from atmosphere blocking and gravity waves, blood vessels, Bose–Einstein condensates, rods and positons and so on. With the aid of symbolic computation, a generalized Miura transformation is proposed to relate the solutions of the vcKdV equation to those of a variable-coefficient modified Korteweg–de Vries equation. Then by using such a Miura transformation and the Galilean invariant transformation, the existence of infinite conservation laws are proved under the Painlevé integrable condition. These results may be valuable for the new discoveries in dynamical systems described by integrable vcKdV models and the theoretical study of the relationships among infinite conservation laws, the integrability of the nonlinear evolution equation and inverse scattering transform.

scholarly journals Exact Explicit Solutions and Conservation Laws for a Coupled Zakharov-Kuznetsov System

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Chaudry Masood Khalique
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
Equation Method ◽  
Explicit Solutions ◽  
Simplest Equation Method ◽  
De Vries

We study a coupled Zakharov-Kuznetsov system, which is an extension of a coupled Korteweg-de Vries system in the sense of the Zakharov-Kuznetsov equation. Firstly, we obtain some exact solutions of the coupled Zakharov-Kuznetsov system using the simplest equation method. Secondly, the conservation laws for the coupled Zakharov-Kuznetsov system will be constructed by using the multiplier approach.

Ablowitz–Kaup–Newell–Segur system, conservation laws and Bäcklund transformation of a variable-coefficient Korteweg–de Vries equation in plasma physics, fluid dynamics or atmospheric science

2020 ◽  
Vol 34 (25) ◽  
pp. 2050226 ◽  
Author(s):  
Yu-Qi Chen ◽  
Bo Tian ◽  
Qi-Xing Qu ◽  
He Li ◽  
Xue-Hui Zhao ◽  
...  
Keyword(s):  
Conservation Laws ◽  
Variable Coefficient ◽  
Vries Equation ◽  
Bäcklund Transformation ◽  
Atmospheric Science ◽  
Backlund Transformation ◽  
Atmospheric Flow ◽  
Bilinear Bäcklund Transformation ◽  
De Vries ◽  
Nonlinearity Dispersion

For a variable-coefficient Korteweg–de Vries equation in a lake/sea, two-layer liquid, atmospheric flow, cylindrical plasma or interactionless plasma, in this paper, we derive the bilinear Bäcklund transformation, non-isospectral Ablowitz–Kaup–Newell–Segur system and infinite conservation laws for the wave amplitude under certain constraints among the external force, dissipation, nonlinearity, dispersion and perturbation.

scholarly journals Conservation Laws for a Degasperis Procesi Equation and a Coupled Variable-Coefficient Modified Korteweg-de Vries System in a Two-Layer Fluid Model via the Multiplier Approach

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
E. Osman ◽  
M. Khalfallah ◽  
H. Sapoor
Keyword(s):  
Conservation Laws ◽  
Variable Coefficient ◽  
Fluid Model ◽  
Variational Derivative ◽  
Derivative Method ◽  
De Vries

We employ the multiplier approach (variational derivative method) to derive the conservation laws for the Degasperis Procesi equation and a coupled variable-coefficient modified Korteweg-de Vries system in a two-layer fluid model. Firstly, the multipliers are computed and then conserved vectors are obtained for each multiplier.

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