scholarly journals KdV Equation Model in Open Cylindrical Channel under Precession

2021 ◽  
Vol 28 (4) ◽  
pp. 466-491
Author(s):  
Hajar Alshoufi
Keyword(s):  
Vries Equation ◽  
Kdv Equation ◽  
Additional Term ◽  
Vertical Axis ◽  
Resonant Mode ◽  
Equation Model ◽  
Concentric Cylinders ◽  
De Vries ◽  
Forced Term ◽  
Good Agreement

AbstractA new model for Korteweg and de-Vries equation (KdV) is derived. The system under study is an open channel consisting of two concentric cylinders, rotating about their vertical axis, which is tilted by slope $$\tau$$ τ from the inertial vertical $$z$$ z , in uniform rate $${\Omega }_{1}=\tau \Omega$$ Ω 1 = τ Ω , and the whole tank is elevated over other table rotating at rate $$\Omega$$ Ω . Under these conditions, a set of Kelvin waves is formed on the free surface depending on the angle of tilt, characterized by the slope $$\tau$$ τ , volume of water, and rotation rate. The resonant mode in the system appears in the form of a single Kelvin solitary wave, whose amplitude satisfies the Korteweg-de Vries equation with forced term. The equation was derived following classical perturbation methods, the additional term made the equation a non-integrable one, that cannot be solved without the help of numerical methods. Invoking the simple finite difference scheme method, it was found that the numerical results are in a good agreement with the experiment.

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.

scholarly journals Integration of the modified Korteweg-de Vries equation with an additional term

2021 ◽  
Author(s):  
Акназар Хасанов ◽  
Умид Хоитметов
Keyword(s):  
Vries Equation ◽  
Additional Term ◽  
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.

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 Approximate Analytical Solution for the Forced Korteweg-de Vries Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Vincent Daniel David ◽  
Mojtaba Nazari ◽  
Vahid Barati ◽  
Faisal Salah ◽  
Zainal Abdul Aziz
Keyword(s):  
Vries Equation ◽  
Nonlinear Problems ◽  
Analysis Method ◽  
Convergence Region ◽  
Analytical Technique ◽  
De Vries ◽  
Homotopy Analysis ◽  
And Control ◽  
Good Agreement ◽  
Fkdv Equation

The forced Korteweg-de Vries (fKdV) equations are solved using Homotopy Analysis Method (HAM). HAM is an approximate analytical technique which provides a novel way to obtain series solutions of such nonlinear problems. It has the auxiliary parameterℏ, where it is easy to adjust and control the convergence region of the series solution. Some examples of forcing terms are employed to analyse the behaviours of the HAM solutions for the different fKdV equations. Finally, this form of HAM solution is compared with the analytical soliton-type solution of fKdV equation as derived by Zhao and Guo. The results is found to be in good agreement with Zhao and Guo.

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.

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