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.

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.

Nonlocal Symmetries, Explicit Solutions, and Wave Structures for the Korteweg–de Vries Equation

2016 ◽  
Vol 71 (8) ◽  
pp. 735-740
Author(s):  
Zheng-Yi Ma ◽  
Jin-Xi Fei
Keyword(s):  
Vries Equation ◽  
Kdv Equation ◽  
Elliptic Functions ◽  
Group Method ◽  
Physical Interaction ◽  
Nonlocal Symmetries ◽  
Interaction Solutions ◽  
The Kdv Equation ◽  
De Vries ◽  
Exact Invariant

AbstractFrom the known Lax pair of the Korteweg–de Vries (KdV) equation, the Lie symmetry group method is successfully applied to find exact invariant solutions for the KdV equation with nonlocal symmetries by introducing two suitable auxiliary variables. Meanwhile, based on the prolonged system, the explicit analytic interaction solutions related to the hyperbolic and Jacobi elliptic functions are derived. Figures show the physical interaction between the cnoidal waves and a solitary wave.

scholarly journals On the Integration of the matrix modified Korteweg-de Vries equation with a self-consistent source

2019 ◽  
Vol 50 (3) ◽  
pp. 281-291 ◽  
Author(s):  
G. U. Urazboev ◽  
A. K. Babadjanova
Keyword(s):  
Vries Equation ◽  
Kdv Equation ◽  
Scattering Data ◽  
Self Consistent ◽  
De Vries ◽  
The Matrix

In this work we deduce laws of the evolution of the scattering  data for the matrix Zakharov Shabat system with the potential that is the solution of the matrix modied KdV equation with a self consistent source.

Wronskian solutions and integrability for a generalized variable-coefficient forced Korteweg–de Vries equation in fluids

2011 ◽  
Vol 67 (2) ◽  
pp. 1023-1030 ◽  
Author(s):  
Xin Yu ◽  
Yi-Tian Gao ◽  
Zhi-Yuan Sun ◽  
Ying Liu
Keyword(s):  
Variable Coefficient ◽  
Vries Equation ◽  
De Vries

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

2019 ◽  
Vol 33 (27) ◽  
pp. 1950319 ◽  
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.

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