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.

scholarly journals Soliton–Breather Interaction: The Modified Korteweg–de Vries Equation Framework

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1445
Author(s):  
Ekaterina Didenkulova ◽  
Efim Pelinovsky
Keyword(s):  
Optical Fibers ◽  
Water Waves ◽  
Relative Phase ◽  
Vries Equation ◽  
Decisive Role ◽  
Pairwise Interactions ◽  
De Vries ◽  
Series Of Experiments ◽  
Stratified Ocean ◽  
The Moment

Pairwise interactions of particle-like waves (such as solitons and breathers) are important elementary processes that play a key role in the formation of the rarefied soliton gas statistics. Such waves appear in different physical systems such as deep water, shallow water waves, internal waves in the stratified ocean, and optical fibers. We study the features of different regimes of collisions between a soliton and a breather in the framework of the focusing modified Korteweg–de Vries equation, where cubic nonlinearity is essential. The relative phase of these structures is an important parameter determining the dynamics of soliton–breather collisions. Two series of experiments with different values of the breather’s and soliton’s relative phases were conducted. The waves’ amplitudes resulting from the interaction of coherent structures depending on their relative phase at the moment of collision were analyzed. Wave field moments, which play a decisive role in the statistics of soliton gases, were determined.

Asymptotics of solutions to a fifth-order modified Korteweg–de Vries equation in the quarter plane

2020 ◽  
pp. 1-46
Author(s):  
Nan Liu ◽  
Boling Guo
Keyword(s):  
Large Time ◽  
Vries Equation ◽  
Large Time Behavior ◽  
Descent Method ◽  
Steepest Descent Method ◽  
Time Behavior ◽  
Transform Method ◽  
Quarter Plane ◽  
De Vries ◽  
Fifth Order

The large-time behavior of solutions to a fifth-order modified Korteweg–de Vries equation in the quarter plane is established. Our approach uses the unified transform method of Fokas and the nonlinear steepest descent method of Deift and Zhou.

Irrotational water waves and the complex Korteweg-de Vries Equation

1996 ◽  
Vol 98 (2-4) ◽  
pp. 510-514 ◽  
Author(s):  
D. Levi ◽  
M. Sanielevici
Keyword(s):  
Water Waves ◽  
Vries Equation ◽  
De Vries

scholarly journals Time Fractional Generalized Korteweg-de Vries Equation: Explicit Series Solutions and Exact Solutions

2021 ◽  
Vol 2 (2) ◽  
pp. 62-77
Author(s):  
Rajeev Kumar ◽  
Sanjeev Kumar ◽  
Sukhneet Kaur ◽  
Shrishty Jain
Keyword(s):  
Differential Equations ◽  
Series Solution ◽  
Vries Equation ◽  
Convergent Series ◽  
De Vries ◽  
Conformable Derivatives ◽  
Graphical View ◽  
Conditionally Convergent Series ◽  
The Given ◽  
Explicit Series Solutions

In this article, an attempt is made to achieve the series solution of the time fractional generalized Korteweg-de Vries equation which leads to a conditionally convergent series solution. We have also resorted to another technique involving conversion of the given fractional partial differential equations to ordinary differential equations by using fractional complex transform. This technique is discussed separately for modified Riemann-Liouville and conformable derivatives. Convergence analysis and graphical view of the obtained solution are demonstrated in this work.

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