scholarly journals Soliton solutions to the fifth-order Korteweg–de Vries equation and their applications to surface and internal water waves

2018 ◽  
Vol 30 (2) ◽  
pp. 022104 ◽  
Author(s):  
K. R. Khusnutdinova ◽  
Y. A. Stepanyants ◽  
M. R. Tranter
Keyword(s):  
Water Waves ◽  
Vries Equation ◽  
Soliton Solutions ◽  
Internal Water ◽  
De Vries ◽  
Fifth Order

scholarly journals Traveling wave solutions for the extended modified KORTEWEG-DE VRIES equation

2021 ◽  
Vol 82 (4) ◽  
pp. 197-203
Author(s):  
G. N. Shaikhova ◽  
◽  
B. K. Rakhimzhanov ◽  
Keyword(s):  
Traveling Wave ◽  
Vries Equation ◽  
Nonlinear Partial Differential Equations ◽  
Integrable Model ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Mathematical Tool ◽  
Wave Solutions ◽  
De Vries ◽  
Fifth Order

In this paper, we study an extended modified Korteweg-de Vries equation, which contains the relevant higher-order nonlinear terms and fifth-order dispersion. This equation is the extension of the modified Korteweg-de Vries equation and described by the Ablowitz-Kaup-Newell-Segur hierarchy. The standard Korteweg-de Vries equation is the pioneer integrable model in solitary waves theory, which gives rise to multiple soliton solutions. The Korteweg-de Vries equation arises naturally from shallow water, plasma physics, and other fields of science. To obtain exact solutions the sine-cosine method is applied. It is shown that the sine-cosine method provides a powerful mathematical tool for solving a great many nonlinear partial differential equations in mathematical physics. Traveling wave solutions are determined for extended modified Korteweg-de Vries equation. The study shows that the sine–cosine method is quite efficient and practically well suited for use in calculating traveling wave solutions for extended modified Korteweg-de Vries equation.

scholarly journals SIMULATION OF KORTEWEG DE VRIES EQUATION

2012 ◽  
Vol 09 ◽  
pp. 574-580
Author(s):  
S. MAT ZIN ◽  
W. N. M. ARIFFIN ◽  
S. A. HASHIM ALI
Keyword(s):  
Mathematical Model ◽  
Shallow Water ◽  
Nonlinear Wave ◽  
Water Waves ◽  
Vries Equation ◽  
Kdv Equation ◽  
Soliton Solutions ◽  
Interaction Process ◽  
Shallow Water Waves ◽  
De Vries

Korteweg de Vries (KdV) equation has been used as a mathematical model of shallow water waves. In this paper, we present one-, two-, and three-soliton solution of KdV equation. By definition, soliton is a nonlinear wave that maintains its properties (shape and velocity) upon interaction with each other. In order to investigate the behavior of soliton solutions of KdV equation and the interaction process of the two- and three-solitons, computer programs have been successfully simulated. Results from these simulations confirm that the solutions of KdV equation obtained are the soliton solutions.

The Korteweg-de Vries equation and water waves. Solutions of the equation. Part 1

1973 ◽  
Vol 59 (4) ◽  
pp. 721-736 ◽  
Author(s):  
Harvey Segur
Keyword(s):  
Water Waves ◽  
Large Time ◽  
Vries Equation ◽  
Asymptotic Properties ◽  
Series Representation ◽  
Convergent Series ◽  
Asymptotic Results ◽  
De Vries ◽  
Method Of Solution

The method of solution of the Korteweg–de Vries equation outlined by Gardneret al.(1967) is exploited to solve the equation. A convergent series representation of the solution is obtained, and previously known aspects of the solution are related to this general form. Asymptotic properties of the solution, valid for large time, are examined. Several simple methods of obtaining approximate asymptotic results are considered.

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