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

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.

N-SOLITON-LIKE SOLUTIONS AND BÄCKLUND TRANSFORMATIONS FOR A NON-ISOSPECTRAL AND VARIABLE-COEFFICIENT MODIFIED KORTEWEG-DE VRIES EQUATION

2011 ◽  
Vol 25 (05) ◽  
pp. 723-733 ◽  
Author(s):  
QIAN FENG ◽  
YI-TIAN GAO ◽  
XIANG-HUA MENG ◽  
XIN YU ◽  
ZHI-YUAN SUN ◽  
...  
Keyword(s):  
Variable Coefficient ◽  
Vries Equation ◽  
Bäcklund Transformation ◽  
Lax Pair ◽  
Mkdv Equation ◽  
Wronskian Technique ◽  
Backlund Transformation ◽  
Bilinear Bäcklund Transformation ◽  
De Vries ◽  
Bilinear Equations

A non-isospectral and variable-coefficient modified Korteweg–de Vries (mKdV) equation is investigated in this paper. Starting from the Ablowitz–Kaup–Newell–Segur procedure, the Lax pair is established and the Bäcklund transformation in original variables is also derived. By a dependent variable transformation, the non-isospectral and variable-coefficient mKdV equation is transformed into bilinear equations, by virtue of which the N-soliton-like solution is obtained. In addition, the bilinear Bäcklund transformation gives a one-soliton-like solution from a vacuum one. Furthermore, the N-soliton-like solution in the Wronskian form is constructed and verified via the Wronskian technique.

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.

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