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

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.

Download Full-text

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

International Journal of Modern Physics B ◽

10.1142/s0217979209049851 ◽

2009 ◽

Vol 23 (04) ◽

pp. 571-584 ◽

Cited By ~ 5

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.

Download Full-text

Lump and Lump-Type Solutions of the Generalized (3+1)-Dimensional Variable-Coefficient B-Type Kadomtsev-Petviashvili Equation

Journal of Applied Mathematics ◽

10.1155/2019/7172860 ◽

2019 ◽

Vol 2019 ◽

pp. 1-5 ◽

Cited By ~ 3

Author(s):

Yanni Zhang ◽

Jing Pang

Keyword(s):

Symbolic Computation ◽

Bilinear Form ◽

Variable Coefficient ◽

Three Dimensional ◽

Petviashvili Equation ◽

Hirota Bilinear Form ◽

Contour Plots ◽

Dimensional Variable ◽

Hirota Bilinear

Based on the Hirota bilinear form of the generalized (3+1)-dimensional variable-coefficient B-type Kadomtsev-Petviashvili equation, the lump and lump-type solutions are generated through symbolic computation, whose analyticity can be easily achieved by taking special choices of the involved parameters. The property of solutions is investigated and exhibited vividly by three-dimensional plots and contour plots.

Download Full-text

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

International Journal of Modern Physics B ◽

10.1142/s0217979209052182 ◽

2009 ◽

Vol 23 (10) ◽

pp. 2383-2393 ◽

Cited By ~ 1

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.

Download Full-text

Lump-type solutions, interaction solutions and periodic wave solutions of a (3+1)-dimensional Korteweg–de Vries equation

International Journal of Modern Physics B ◽

10.1142/s0217979219503193 ◽

2019 ◽

Vol 33 (27) ◽

pp. 1950319 ◽

Cited By ~ 1

Author(s):

Hongfei Tian ◽

Jinting Ha ◽

Huiqun Zhang

Keyword(s):

Vries Equation ◽

Kdv Equation ◽

Periodic Wave ◽

Periodic Wave Solutions ◽

Interaction Solutions ◽

Dynamical Behaviors ◽

Wave Solutions ◽

Hirota Bilinear Form ◽

De Vries ◽

Hirota Bilinear

Based on the Hirota bilinear form, lump-type solutions, interaction solutions and periodic wave solutions of a (3[Formula: see text]+[Formula: see text]1)-dimensional Korteweg–de Vries (KdV) equation are obtained. The interaction between a lump-type soliton and a stripe soliton including two phenomena: fission and fusion, are illustrated. The dynamical behaviors are shown more intuitively by graphics.

Download Full-text

Solitons for a forced generalized variable-coefficient Korteweg–de Vries equation for the atmospheric blocking phenomenon

Modern Physics Letters B ◽

10.1142/s0217984916503188 ◽

2016 ◽

Vol 30 (35) ◽

pp. 1650318 ◽

Cited By ~ 1

Author(s):

Jun Chai ◽

Bo Tian ◽

Xi-Yang Xie ◽

Han-Peng Chai

Keyword(s):

Variable Coefficient ◽

Vries Equation ◽

Soliton Solutions ◽

Atmospheric Blocking ◽

Graphic Analysis ◽

Bilinear Bäcklund Transformation ◽

Different Types ◽

De Vries ◽

Soliton Amplitude ◽

The One

Investigation is given to a forced generalized variable-coefficient Korteweg–de Vries equation for the atmospheric blocking phenomenon. Applying the double-logarithmic and rational transformations, respectively, under certain variable-coefficient constraints, we get two different types of bilinear forms: (a) Based on the first type, the bilinear Bäcklund transformation (BT) is derived, the [Formula: see text]-soliton solutions in the Wronskian form are constructed, and the [Formula: see text]- and [Formula: see text]-soliton solutions are proved to satisfy the bilinear BT; (b) Based on the second type, via the Hirota method, the one- and two-soliton solutions are obtained. Those two types of solutions are different. Graphic analysis on the two types shows that the soliton velocity depends on [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text], the soliton amplitude is merely related to [Formula: see text], and the background depends on [Formula: see text] and [Formula: see text], where [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] are the dissipative, dispersive, nonuniform and line-damping coefficients, respectively, and [Formula: see text] is the external-force term. We present some types of interactions between the two solitons, including the head-on and overtaking interactions, interactions between the velocity- and amplitude-unvarying two solitons, between the velocity-varying while amplitude-unvarying two solitons and between the velocity- and amplitude-varying two solitons, as well as the interactions occurring on the constant and varying backgrounds.

Download Full-text

INTEGRABLE PROPERTIES FOR A GENERALIZED NON-ISOSPECTRAL AND VARIABLE-COEFFICIENT KORTEWEG–DE VRIES MODEL

Modern Physics Letters B ◽

10.1142/s0217984910022949 ◽

2010 ◽

Vol 24 (10) ◽

pp. 1023-1032 ◽

Cited By ~ 1

Author(s):

XIAO-GE XU ◽

XIANG-HUA MENG ◽

FU-WEI SUN ◽

YI-TIAN GAO

Keyword(s):

Bilinear Form ◽

Variable Coefficient ◽

Bäcklund Transformation ◽

Backlund Transformation ◽

Bilinear Method ◽

Solitonic Solution ◽

Bilinear Bäcklund Transformation ◽

De Vries ◽

The One ◽

Hirota Bilinear

Applicable in fluid dynamics and plasmas, a generalized variable-coefficient Korteweg–de Vries (vcKdV) equation is investigated analytically employing the Hirota bilinear method in this paper. The bilinear form for such a model is derived through a dependent variable transformation. Based on the bilinear form, the integrable properties such as the N-solitonic solution, the Bäcklund transformation and the Lax pair for the vcKdV equation are obtained. Additionally, it is shown that the bilinear Bäcklund transformation can turn into the one denoted in the original variables.

Download Full-text