Reduction and analytic solutions of a variable-coefficient Korteweg–de Vries equation in a fluid, crystal or plasma

2020 ◽  
Vol 34 (26) ◽  
pp. 2050287 ◽  
Author(s):  
Yu-Qi Chen ◽  
Bo Tian ◽  
Qi-Xing Qu ◽  
He Li ◽  
Xue-Hui Zhao ◽  
...  
Keyword(s):  
Variable Coefficient ◽  
Vries Equation ◽  
Kdv Equation ◽  
Circulatory System ◽  
Analytic Solutions ◽  
Original Equation ◽  
Water Tunnel ◽  
Anharmonic Crystal ◽  
Velocity Amplitude ◽  
Mixed Solutions

In this paper, a variable-coefficient KdV equation in a fluid, plasma, anharmonic crystal, blood vessel, circulatory system, shallow-water tunnel, lake or relaxation inhomogeneous medium is discussed. We construct the reduction from the original equation to another variable-coefficient KdV equation, and then get the rational, periodic and mixed solutions of the original equation under certain constraint. For the original equation, we obtain that (i) the dispersive coefficient affects the solitonic background, velocity and amplitude; (ii) the perturbed coefficient affects the solitonic velocity, amplitude and background; (iii) the dissipative coefficient affects the solitonic background, and there are different mixed solutions under the same constraint with the dispersive, perturbed and dissipative coefficients changing.

BÄCKLUND TRANSFORMATION AND ANALYTIC SOLUTIONS FOR A GENERALIZED VARIABLE-COEFFICIENT MODIFIED KORTEWEG–DE VRIES MODEL FROM FLUID DYNAMICS AND PLASMAS

2008 ◽  
Vol 19 (11) ◽  
pp. 1659-1671 ◽  
Author(s):  
FU-WEI SUN ◽  
YI-TIAN GAO ◽  
CHUN-YI ZHANG ◽  
XIAO-GE XU
Keyword(s):  
Fluid Dynamics ◽  
External Force ◽  
Variable Coefficient ◽  
Bäcklund Transformation ◽  
Solitary Wave Solution ◽  
Kdv Equation ◽  
Analytic Solutions ◽  
Mkdv Equation ◽  
Backlund Transformation ◽  
De Vries

We investigate a generalized variable-coefficient modified Korteweg–de Vries model with perturbed factor and external force (vc-GmKdV) describing fluid dynamics and space plasmas. In this paper, we propose an extended variable-coefficient balancing-act method (Evc-BAM), which is concise and straightforward, to obtain the generalized analytic solutions including solitary wave solution of the vc-GmKdV model with symbolic computation. Meanwhile, using the Evc-BAM, we obtain an auto-Bäcklund transformation for the vc-GmKdV model on the relevant constraint conditions of the coefficient functions. Using the given auto-Bäcklund transformation, the solutions of special equations for the vc-GmKdV model are also obtained as the variable-coefficient Korteweg–de Vries (vc-KdV) equation, the generalized KdV equation with perturbed factor and external force (GKdV), the variable-coefficient modified Korteweg–de Vries (vc-mKdV) equation, and the variable-coefficient cylindrical modified Korteweg–de Vries (vc-cmKdV) equation, respectively.

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.

POSITONIC SOLUTIONS OF A VARIABLE-COEFFICIENT KdV EQUATION IN FLUID DYNAMICS AND PLASMA PHYSICS

2001 ◽  
Vol 12 (10) ◽  
pp. 1431-1437 ◽  
Author(s):  
YI-TIAN GAO ◽  
BO TIAN
Keyword(s):  
Fluid Dynamics ◽  
Plasma Physics ◽  
Variable Coefficient ◽  
Kdv Equation ◽  
Analytic Solutions ◽  
Exact Analytic Solutions ◽  
De Vries

Positons are a new concept in nonlinear sciences. In this paper, we report some positonic, exact analytic solutions for a variable-coefficient Korteweg-de Vries (KdV) equation arising from fluid dynamics and plasma physics.

Periodic and decay mode solutions of the generalized variable-coefficient Korteweg–de Vries equation

2019 ◽  
Vol 33 (20) ◽  
pp. 1950234 ◽  
Author(s):  
Yufeng Zhang ◽  
Jiangen Liu
Keyword(s):  
Periodic Solutions ◽  
Decay Mode ◽  
Variable Coefficient ◽  
Vries Equation ◽  
Kdv Equation ◽  
Variation Of Parameters ◽  
Triply Periodic ◽  
De Vries ◽  
Mode Solution ◽  
First Time

In this paper, we can obtain periodic and decay mode solutions of the generalized variable-coefficient Korteweg–de Vries (gvc-KdV) equation for the first time by using the multivariate transformation technique. Periodic solutions are doubly and triply periodic solutions, and the decay mode solutions include the 1-decay solution and the 2-decay mode solution. Furthermore, we will also study the effect of variation of parameters for solutions systematically, relating to different forms of expression, through graphic display.

scholarly journals Wave Breaking in Undular Bores with Shear Flows

Water Waves ◽  
2021 ◽  
Author(s):  
Maria Bjørnestad ◽  
Henrik Kalisch ◽  
Malek Abid ◽  
Christian Kharif ◽  
Mats Brun
Keyword(s):  
Turbulent Flows ◽  
Wave Breaking ◽  
Vries Equation ◽  
Kdv Equation ◽  
Flow Depth ◽  
Present Contribution ◽  
Undular Bore ◽  
Wave Flume ◽  
The Kdv Equation ◽  
The Waves

AbstractIt is well known that weak hydraulic jumps and bores develop a growing number of surface oscillations behind the bore front. Defining the bore strength as the ratio of the head of the undular bore to the undisturbed depth, it was found in the classic work of Favre (Ondes de Translation. Dunod, Paris, 1935) that the regime of laminar flow is demarcated from the regime of partially turbulent flows by a sharply defined value 0.281. This critical bore strength is characterized by the eventual breaking of the leading wave of the bore front. Compared to the flow depth in the wave flume, the waves developing behind the bore front are long and of small amplitude, and it can be shown that the situation can be described approximately using the well known Kortweg–de Vries equation. In the present contribution, it is shown that if a shear flow is incorporated into the KdV equation, and a kinematic breaking criterion is used to test whether the waves are spilling, then the critical bore strength can be found theoretically within an error of less than ten percent.

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