The Korteweg-de Vries Equation (KdV-Equation)

1981 ◽  
pp. 12-26
Author(s):  
Gert Eilenberger
Keyword(s):  
Vries Equation ◽  
Kdv Equation ◽  
De Vries

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.

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.

Decay of the solutions of the generalized Korteweg–de Vries equation at large times

2019 ◽  
pp. 99-111
Author(s):  
Artyom Nikolayev
Keyword(s):  
Weak Solutions ◽  
Vries Equation ◽  
Kdv Equation ◽  
Existence Of Weak Solutions ◽  
Generalized Kdv Equation ◽  
De Vries

In this paper the existence of weak solutions of the nonlinear generalized KdV equation is shown and conditions for which weak solutions decay to zero at large times are obtained.

scholarly journals On the semi-global stabilizability of the Korteweg-de Vries Equation via model predictive control

2018 ◽  
Vol 24 (1) ◽  
pp. 237-263 ◽  
Author(s):  
Behzad Azmi ◽  
Anne-Céline Boulanger ◽  
Karl Kunisch
Keyword(s):  
Model Predictive Control ◽  
Numerical Experiment ◽  
Exponential Stability ◽  
Predictive Control ◽  
Vries Equation ◽  
Kdv Equation ◽  
Bounded Interval ◽  
De Vries ◽  
The Stability ◽  
Theoretical Results

Stabilization of the nonlinear Korteweg-de Vries (KdV) equation on a bounded interval by model predictive control (MPC) is investigated. This MPC strategy does not need any terminal cost or terminal constraint to guarantee the stability. The semi-global stabilizability is the key condition. Based on this condition, the suboptimality and exponential stability of the model predictive control are investigated. Finally, numerical experiment is presented which validates the theoretical results.

scholarly journals Exact Solitary Wave Solutions for the Generalized Schamel Korteweg–De Vries Equation

2017 ◽  
Vol 9 (5) ◽  
pp. 126 ◽  
Author(s):  
N. O. Al Atawi
Keyword(s):  
Acoustic Waves ◽  
Vries Equation ◽  
Kdv Equation ◽  
Nonlinear Partial Differential Equations ◽  
Mathematical Tool ◽  
Ion Acoustic Waves ◽  
Physical Problems ◽  
De Vries ◽  
Powerful Mathematical Tool ◽  
Important Significance

The generalized Schamel-Korteweg-de Vries (S-KdV) equation containing root of degree nonlinearity is a very attractive model for the study of ion-acoustic waves in plasma and dusty plasma. In this work, we obtain the soliton-like solutions, the kink solutions, and the plural solutions of the generalized S-KdV equation by using the sine-cosine method. These solutions may be of important significance for the explanation of some practical physical problems. It is shown that these two methods provide a powerful mathematical tool for solving a great many nonlinear partial differential equations in mathematical physics.

Non-local symmetries and the nth finite symmetry transformations for the (2+1)-dimensional Korteweg–de Vries equation

2020 ◽  
pp. 2050432
Author(s):  
Xiazhi Hao ◽  
Xiaoyan Li
Keyword(s):  
Spectral Function ◽  
Soliton Solution ◽  
Vries Equation ◽  
Kdv Equation ◽  
Symmetry Transformation ◽  
De Vries ◽  
Symmetry Transformations ◽  
Non Local ◽  
Local Symmetries

Non-local symmetries in forms of square spectral function and residue over the (2+1)-dimensional Korteweg–de Vries (KdV) equation are studied in some detail. Then, we present [Formula: see text]-soliton solution to this equation with the help of symmetry transformation.

LOW-DIMENSIONAL CHAOS IN A PERTURBED KORTEWEG–DE VRIES EQUATION

1995 ◽  
Vol 05 (04) ◽  
pp. 1221-1233 ◽  
Author(s):  
X. TIAN ◽  
R. H. J. GRIMSHAW
Keyword(s):  
Steady State ◽  
Vries Equation ◽  
Numerical Study ◽  
Kdv Equation ◽  
Period Doubling ◽  
Subharmonic Bifurcation ◽  
De Vries ◽  
Route To Chaos ◽  
Low Dimensional ◽  
Perturbed Kdv Equation

Spatial chaos has been observed in the steady state from a numerical study of a perturbed Korteweg–de Vries equation. The onset of chaos is due to a subharmonic bifurcation sequence. A second route to chaos is also observed via a period-doubling sequence generated from each fundamental subharmonic state. In this paper, the question of determining low-dimensional chaos in this perturbed KdV equation is addressed. The dimension of this system in the steady state is estimated from the corresponding ordinary differential equation via the Lyapunov spectrum, and also from a numerical investigation via a reconstructed attractor using a spatial series.

scholarly journals On the Korteweg-de Vries equation: an associated equation

1984 ◽  
Vol 7 (2) ◽  
pp. 263-277 ◽  
Author(s):  
Eugene P. Schlereth ◽  
Ervin Y. Rodin
Keyword(s):  
Initial Value Problems ◽  
Vries Equation ◽  
Kdv Equation ◽  
Nonlinear Partial Differential Equation ◽  
Inverse Scattering Transform ◽  
Similarity Solutions ◽  
Airy Functions ◽  
Initial Value ◽  
Scattering Transform ◽  
De Vries

The purpose of this paper is to describe a relationship between the Korteweg-de Vries (KdV) equationut−6uux+uxxx=0and another nonlinear partial differential equation of the formzt+zxxx−3zxzxxz=H(t)z.The second equation will be called the Associated Equation (AE) and the connection between the two will be explained. By considering AE, explicit solutions to KdV will be obtained. These solutions include the solitary wave and the cnoidal wave solutions. In addition, similarity solutions in terms of Airy functions and Painlevé transcendents are found. The approach here is different from the Inverse Scattering Transform and the results are not in the form of solutions to specific initial value problems, but rather in terms of solutions containing arbitrary constants.

Analytic Solitary-wave Solutions for Modified Korteweg - de Vries Equation with t-dependent Coefficients

2001 ◽  
Vol 56 (5) ◽  
pp. 366-370 ◽  
Author(s):  
Woo-Pyo Hong ◽  
Myung-Sang Yoona
Keyword(s):  
Solitary Wave ◽  
Symbolic Computation ◽  
Vries Equation ◽  
Bäcklund Transformation ◽  
Kdv Equation ◽  
Solitary Wave Solutions ◽  
Wave Solutions ◽  
Modified Kdv Equation ◽  
De Vries ◽  
Truncated Painlevé Expansion

Abstract We find analytic solitary wave solutions for a modified KdV equation with t-dependent coefficients of the form ut - 6α(t)uux + ß (t) uxxx -6γu2ux = 0. We make use of both the application of the truncated Painleve expansion and symbolic computation to obtain an auto-Bäcklund transformation. We show that kink-type analytic solitary-wave solutions exist under some constraints on α (t), ß (t) and γ.

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