On the integrability properties of variable coefficient Korteweg – de Vries equations

1996 ◽  
Vol 74 (9-10) ◽  
pp. 676-684 ◽  
Author(s):  
F. Güngör ◽  
M. Sanielevici ◽  
P. Winternitz
Keyword(s):  
Differential Equations ◽  
Variable Coefficient ◽  
Kdv Equation ◽  
Three Dimensional ◽  
Symmetry Algebra ◽  
Time Symmetry ◽  
Kdv Equations ◽  
Painlevé Property ◽  
The Kdv Equation ◽  
De Vries

All variable coefficient Korteweg – de Vries (KdV) equations with three-dimensional Lie point symmetry groups are investigated. For such an equation to have the Painlevé property, its coefficients must satisfy seven independent partial differential equations. All of them are satisfied only for equations equivalent to the KdV equation itself. However, most of them are satisfied in all cases. If the symmetry algebra is either simple, or nilpotent, then the equations have families of single-valued solutions depending on two arbitrary functions of time. Symmetry reduction is used to obtain particular solutions. The reduced ordinary differential equations are classified.

An index theorem for the stability of periodic travelling waves of Korteweg–de Vries type

2011 ◽  
Vol 141 (6) ◽  
pp. 1141-1173 ◽  
Author(s):  
Jared C. Bronski ◽  
Mathew A. Johnson ◽  
Todd Kapitula
Keyword(s):  
Blow Up ◽  
Kdv Equation ◽  
Travelling Waves ◽  
Classical Mechanics ◽  
Index Theorem ◽  
Travelling Wave ◽  
Kdv Equations ◽  
The Kdv Equation ◽  
De Vries ◽  
The Stability

We consider the stability of periodic travelling-wave solutions to a generalized Korteweg–de Vries (gKdV) equation and prove an index theorem relating the number of unstable and potentially unstable eigenvalues to geometric information on the classical mechanics of the travelling-wave ordinary differential equation. We illustrate this result with several examples, including the integrable KdV and modified KdV equations, the L2-critical KdV-4 equation that arises in the study of blow-up and the KdV-½ equation, which is an idealized model for plasmas.

scholarly journals Nonlinear water waves (KdV) equation and Painleve technique

2015 ◽  
Vol 4 (2) ◽  
pp. 216
Author(s):  
Attia Mostafa
Keyword(s):  
Exact Solutions ◽  
Water Waves ◽  
Kdv Equation ◽  
Nonlinear Pde ◽  
Nonlinear Water Waves ◽  
Third Order ◽  
The Third ◽  
Painlevé Property ◽  
The Kdv Equation ◽  
De Vries

<p>The Korteweg-de Vries (KdV) equation which is the third order nonlinear PDE has been of interest since Scott Russell (1844) . In this paper we study this kind of equation by Painleve equation and through this study, we find that KdV equation satisfies Painleve property, but we could not find a solution directly, so we transformed the KdV equation to the like-KdV equation, therefore, we were able to find four exact solutions to the original KdV equation.</p>

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 Solutions and Painlevé Property for the KdV Equation with Self-Consistent Source

2018 ◽  
Vol 2018 ◽  
pp. 1-16
Author(s):  
Yali Shen ◽  
Ruoxia Yao
Keyword(s):  
Kdv Equation ◽  
Traveling Wave Solutions ◽  
Elliptic Functions ◽  
Expansion Method ◽  
Hyperbolic Function ◽  
Group Invariant ◽  
Painlevé Property ◽  
The Kdv Equation ◽  
Self Consistent ◽  
Group Invariant Solutions

In this paper, the polynomial solutions in terms of Jacobi’s elliptic functions of the KdV equation with a self-consistent source (KdV-SCS) are presented. The extended (G′/G)-expansion method is utilized to obtain exact traveling wave solutions of the KdV-SCS, which finally are expressed in terms of the hyperbolic function, the trigonometric function, and the rational function. Meanwhile we find the Lie point symmetry and Lie symmetry group and give several group-invariant solutions for the KdV-SCS. Finally, we supplement the results of the Painlevé property in our previous work and get the Bäcklund transformations of the KdV-SCS.

scholarly journals Bäcklund transformations for several cases of a type of generalized KdV equation

2004 ◽  
Vol 2004 (63) ◽  
pp. 3369-3377
Author(s):  
Paul Bracken
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Bäcklund Transformation ◽  
Kdv Equation ◽  
Bäcklund Transformations ◽  
Backlund Transformation ◽  
Generalized Kdv Equation ◽  
Backlund Transformations ◽  
De Vries ◽  
Derivative Term

An alternate generalized Korteweg-de Vries system is studied here. A procedure for generating solutions is given. A theorem is presented, which is subsequently applied to this equation to obtain a type of Bäcklund transformation for several specific cases of the power of the derivative term appearing in the equation. In the process, several interesting, new, ordinary, differential equations are generated and studied.

SPHERICAL KORTEWEG – DE VRIES SOLITONS FROM RELATIVISTIC HYDRODYNAMICS

2007 ◽  
Vol 16 (09) ◽  
pp. 3019-3023 ◽  
Author(s):  
D. A. FOGAÇA ◽  
F. S. NAVARRA
Keyword(s):  
Fluid Dynamics ◽  
Equation Of State ◽  
Kdv Equation ◽  
Relativistic Hydrodynamics ◽  
Spherical Coordinates ◽  
Relativistic Fluid Dynamics ◽  
Continuity Equations ◽  
Relativistic Fluid ◽  
The Kdv Equation ◽  
De Vries

We study the conditions for the formation and propagation of Korteweg – de Vries (KdV) solitons in relativistic fluid dynamics using an appropriate equation of state. It is known that in cartesian coordinates the combination of the Euler and continuity equations leads to the KdV equation. Recently it has been shown that this is true also in relativistic hydrodynamics. Here we show that a KdV-like equation can be obtained in relativistic hydrodynamics in spherical coordinates.

scholarly journals A Novel Analytical View of Time-Fractional Korteweg-De Vries Equations via a New Integral Transform

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1254
Author(s):  
Saima Rashid ◽  
Aasma Khalid ◽  
Sobia Sultana ◽  
Zakia Hammouch ◽  
Rasool Shah ◽  
...  
Keyword(s):  
Nonlinear Waves ◽  
Analytical Procedure ◽  
Integral Transform ◽  
Kdv Equation ◽  
Analytical Techniques ◽  
Transform Method ◽  
Fractional Pdes ◽  
Approximate Analytical Solutions ◽  
The Kdv Equation ◽  
De Vries

We put into practice relatively new analytical techniques, the Shehu decomposition method and the Shehu iterative transform method, for solving the nonlinear fractional coupled Korteweg-de Vries (KdV) equation. The KdV equation has been developed to represent a broad spectrum of physics behaviors of the evolution and association of nonlinear waves. Approximate-analytical solutions are presented in the form of a series with simple and straightforward components, and some aspects show an appropriate dependence on the values of the fractional-order derivatives that are, in a certain sense, symmetric. The fractional derivative is proposed in the Caputo sense. The uniqueness and convergence analysis is carried out. To comprehend the analytical procedure of both methods, three test examples are provided for the analytical results of the time-fractional KdV equation. Additionally, the efficiency of the mentioned procedures and the reduction in calculations provide broader applicability. It is also illustrated that the findings of the current methodology are in close harmony with the exact solutions. It is worth mentioning that the proposed methods are powerful and are some of the best procedures to tackle nonlinear fractional PDEs.

Sign in / Sign up

Export Citation Format

Share Document