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.

SIMILARITY REDUCTIONS FOR A GENERALIZED VARIABLE-COEFFICIENT KADOMTSEV–PETVIASHVILI EQUATION WITH SYMBOLIC COMPUTATION

1999 ◽  
Vol 10 (06) ◽  
pp. 1089-1097 ◽  
Author(s):  
BO TIAN
Keyword(s):  
Symbolic Computation ◽  
Water Waves ◽  
Painlevé Equation ◽  
Kp Equation ◽  
Azimuthal Dependence ◽  
Weierstrass Elliptic Function ◽  
Petviashvili Equation ◽  
Painleve Equation ◽  
Second Painlevé Equation ◽  
Similarity Reductions

We use computerized symbolic computation to study the similarity reductions of a generalized Kadomtsev–Petviashvili (KP) equation, with its space- and time-dependent coefficients arising in plasma physics and fluid mechanics. We obtain a couple of reductions involving those coefficient functions, one of which is to the fourth Painlevé equation, while the other is to one of the second Painlevé equation, the first Painlevé equation or a Weierstrass elliptic function equation. Our results agree with the Painlevé Conjecture. The cylindrical KP equation is presented as an example, which describes nonlinear cylindrical water waves in shallow water with a weak azimuthal dependence.

Research Note on New Similarity Reductions of a Variable-Coefficient Korteweg-de Vries Equation

1997 ◽  
Vol 52 (5) ◽  
pp. 463-464
Author(s):  
Bo Tian
Keyword(s):  
Elliptic Function ◽  
Variable Coefficient ◽  
Vries Equation ◽  
Research Note ◽  
Weierstrass Elliptic Function ◽  
De Vries ◽  
Similarity Reductions

Abstract For a variable-coefficient Korteweg-de Vries equation we obtain 4 new similarity reductions to the Painleve type equations or the Weierstrass elliptic function equation.

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.

VARIABLE-COEFFICIENT MIURA TRANSFORMATIONS AND INTEGRABLE PROPERTIES FOR A GENERALIZED VARIABLE-COEFFICIENT KORTEWEG–de VRIES EQUATION FROM BOSE–EINSTEIN CONDENSATES WITH SYMBOLIC COMPUTATION

2009 ◽  
Vol 23 (04) ◽  
pp. 571-584 ◽  
Author(s):  
JUAN LI ◽  
BO TIAN ◽  
XIANG-HUA MENG ◽  
TAO XU ◽  
CHUN-YI ZHANG ◽  
...  
Keyword(s):  
Symbolic Computation ◽  
Variable Coefficient ◽  
Vries Equation ◽  
Kdv Equation ◽  
Nonlinear Superposition ◽  
Modified Kdv Equation ◽  
De Vries ◽  
The One ◽  
Bose Einstein ◽  
Miura Transformations

In this paper, a generalized variable-coefficient Korteweg–de Vries (KdV) equation with the dissipative and/or perturbed/external-force terms is investigated, which arises in arterial mechanics, blood vessels, Bose gases of impenetrable bosons and trapped Bose–Einstein condensates. With the computerized symbolic computation, two variable-coefficient Miura transformations are constructed from such a model to the modified KdV equation under the corresponding constraints on the coefficient functions. Meanwhile, through these two transformations, a couple of auto-Bäcklund transformations, nonlinear superposition formulas and Lax pairs are obtained with the relevant constraints. Furthermore, the one- and two-solitonic solutions of this equation are explicitly presented and the physical properties and possible applications in some fields of these solitonic structures are discussed and pointed out.

ANALYTIC ANALYSIS ON A GENERALIZED (2+1)-DIMENSIONAL VARIABLE-COEFFICIENT KORTEWEG–DE VRIES EQUATION USING SYMBOLIC COMPUTATION

2010 ◽  
Vol 24 (27) ◽  
pp. 5359-5370 ◽  
Author(s):  
CHENG ZHANG ◽  
BO TIAN ◽  
LI-LI LI ◽  
TAO XU
Keyword(s):  
Symbolic Computation ◽  
Variable Coefficient ◽  
Vries Equation ◽  
Nonlinear Superposition ◽  
Hirota Bilinear Form ◽  
De Vries ◽  
The One ◽  
Dimensional Variable ◽  
Hirota Bilinear ◽  
Nonlinear Superposition Formula

With the help of symbolic computation, a generalized (2+1)-dimensional variable-coefficient Korteweg–de Vries equation is studied for its Painlevé integrability. Then, Hirota bilinear form is derived, from which the one- and two-solitary-wave solutions with the corresponding graphic illustration are presented. Furthermore, a bilinear auto-Bäcklund transformation is constructed and the nonlinear superposition formula and Lax pair are also obtained. Finally, the analytic solution in the Wronskian form is constructed and proved by direct substitution into the bilinear equation.

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