Existence of Formal Conservation Laws of a Variable-Coefficient Korteweg–de Vries Equation from Fluid Dynamics and Plasma Physics via Symbolic Computation

2008 ◽  
Vol 25 (3) ◽  
pp. 878-880 ◽  
Author(s):  
Zhang Chun-Yi ◽  
Li Juan ◽  
Meng Xiang-Hua ◽  
Xu Tao ◽  
Gao Yi-Tian
Keyword(s):  
Fluid Dynamics ◽  
Plasma Physics ◽  
Conservation Laws ◽  
Symbolic Computation ◽  
Variable Coefficient ◽  
Vries Equation ◽  
De Vries

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.

Lax Pair and Darboux Transformation for a Variable-Coefficient Fifth-Order Korteweg-de Vries Equation with Symbolic Computation

2008 ◽  
Vol 49 (4) ◽  
pp. 833-838 ◽  
Author(s):  
Zhang Ya-Xing ◽  
Zhang Hai-Qiang ◽  
Li Juan ◽  
Xu Tao ◽  
Zhang Chun-Yi ◽  
...  
Keyword(s):  
Symbolic Computation ◽  
Darboux Transformation ◽  
Variable Coefficient ◽  
Vries Equation ◽  
Lax Pair ◽  
De Vries ◽  
Fifth Order

COMPUTERIZED SYMBOLIC COMPUTATION FOR THE CYLINDRICAL KORTEWEG–DE VRIES EQUATION

1999 ◽  
Vol 10 (07) ◽  
pp. 1303-1316 ◽  
Author(s):  
YI-TIAN GAO ◽  
BO TIAN
Keyword(s):  
Symbolic Computation ◽  
Variable Coefficient ◽  
Vries Equation ◽  
Rational Solution ◽  
Painlevé Equation ◽  
Weierstrass Elliptic Function ◽  
De Vries ◽  
Painleve Equation ◽  
Second Painlevé Equation ◽  
Similarity Reductions

Computers have a great potential in the analytical investigations on various physics problems. In this paper, we make use of computerized symbolic computation to obtain two similarity reductions as well as a rational solution for the variable-coefficient cylindrical Korteweg–de Vries equation, which was originally introduced in the studies of plasma physics. One of the reductions is to the second Painlevé equation, while the other to either the first Painlevé equation or the Weierstrass elliptic function equation. Our results are in agreement with the Painlevé conjecture.

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.

N-SOLITON SOLUTIONS, AUTO-BÄCKLUND TRANSFORMATIONS AND LAX PAIR FOR A NONISOSPECTRAL AND VARIABLE-COEFFICIENT KORTEWEG-DE VRIES EQUATION VIA SYMBOLIC COMPUTATION

2009 ◽  
Vol 23 (10) ◽  
pp. 2383-2393 ◽  
Author(s):  
LI-LI LI ◽  
BO TIAN ◽  
CHUN-YI ZHANG ◽  
HAI-QIANG ZHANG ◽  
JUAN LI ◽  
...  
Keyword(s):  
Symbolic Computation ◽  
Bilinear Form ◽  
Variable Coefficient ◽  
Vries Equation ◽  
Soliton Solutions ◽  
Lax Pair ◽  
Bäcklund Transformations ◽  
Nonlinear Superposition ◽  
Backlund Transformations ◽  
De Vries

In this paper, a nonisospectral and variable-coefficient Korteweg-de Vries equation is investigated based on the ideas of the variable-coefficient balancing-act method and Hirota method. Via symbolic computation, we obtain the analytic N-soliton solutions, variable-coefficient bilinear form, auto-Bäcklund transformations (in both the bilinear form and Lax pair form), Lax pair and nonlinear superposition formula for such an equation in explicit form. Moreover, some figures are plotted to analyze the effects of the variable coefficients on the stabilities and propagation characteristics of the solitonic waves.

scholarly journals Painleve analysis for a forced Korteveg-de Vries equation arisen in fluid dynamics of internal solitary waves

2015 ◽  
Vol 19 (4) ◽  
pp. 1223-1226 ◽  
Author(s):  
Sheng Zhang ◽  
Mei-Tong Chen ◽  
Wei-Yi Qian
Keyword(s):  
Fluid Dynamics ◽  
Solitary Waves ◽  
Variable Coefficient ◽  
Vries Equation ◽  
Internal Solitary Waves ◽  
Painlevé Analysis ◽  
New Exact Solutions ◽  
De Vries ◽  
Painleve Analysis ◽  
Truncated Painlevé Expansion

In this paper, Painleve analysis is used to test the Painleve integrability of a forced variable-coefficient extended Korteveg-de Vries equation which can describe the weakly-non-linear long internal solitary waves in the fluid with continuous stratification on density. The obtained results show that the equation is integrable under certain conditions. By virtue of the truncated Painleve expansion, a pair of new exact solutions to the equation is obtained.

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