Exact solutions of a fifth - order Korteweg–de Vries –type equation modeling nonlinear long waves in several natural phenomena

2021 ◽  
Author(s):  
Elena V. Nikolova ◽  
Mila Chilikova-Lubomirova ◽  
Nikolay K. Vitanov
Keyword(s):  
Exact Solutions ◽  
Type Equation ◽  
Long Waves ◽  
Equation Modeling ◽  
Natural Phenomena ◽  
De Vries ◽  
Fifth Order

A Study for Obtaining more Compacton Solutions of the Modified Form of Fifth-order Korteweg-De Vries-like Equations

2004 ◽  
Vol 59 (6) ◽  
pp. 359-367
Author(s):  
Mustafa Inc
Keyword(s):  
Exact Solutions ◽  
Direct Methods ◽  
Periodic Wave ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
De Vries ◽  
Fifth Order

In this paper we investigate exact solutions of a modified form of fifth-order Korteweg-de Vrieslike equations by using two direct methods. Thus we get new compacton solutions having infinite wings or tails. In addition, new periodic and singular periodic wave solutions are obtained.

Auto-B¨Acklund Transformations And Exact Solutions For The Generalized Two-Dimensional Korteweg-De Vries-Burgers-Type Equations And Burgers-Type Equations

2003 ◽  
Vol 58 (7-8) ◽  
pp. 464-472
Author(s):  
Biao Li ◽  
Yong Chen ◽  
Hongqing Zhang
Keyword(s):  
Exact Solutions ◽  
Bäcklund Transformation ◽  
Type Equation ◽  
Balance Method ◽  
Two Dimensional ◽  
Backlund Transformation ◽  
De Vries ◽  
Homogeneous Balance Method ◽  
Burgers Type Equations ◽  
Homogeneous Balance

In this paper, based on the idea of the homogeneous balance method and with the help of Mathematica, we obtain a new auto-Bäcklund transformation for the generalized two-dimensional Kortewegde Vries-Burgers-type equation and a new auto-Bäcklund transformation for the generalized twodimensional Burgers-type equation by introducing two appropriate transformations. Then, based on these two auto-Bäcklund transformation, some exact solutions for these equations are derived. Some figures are given to show the properties of the solutions.

Quasi-periodic wave solutions and asymptotic properties for a fifth-order Korteweg–de Vries type equation

2016 ◽  
Vol 30 (18) ◽  
pp. 1650223 ◽  
Author(s):  
Chun-Yan Qin ◽  
Shou-Fu Tian ◽  
Xiu-Bin Wang ◽  
Tian-Tian Zhang
Keyword(s):  
Asymptotic Properties ◽  
Soliton Solutions ◽  
Type Equation ◽  
Periodic Wave ◽  
Ocean Dynamics ◽  
Nonlinear Phenomena ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
De Vries ◽  
Fifth Order

Under investigation in this paper is a fifth-order Korteweg–de Vries type (fKdV-type) equation with time-dependent coefficients, which can be used to describe many nonlinear phenomena in fluid mechanics, ocean dynamics and plasma physics. The binary Bell polynomials are employed to find its Hirota’s bilinear formalism with an extra auxiliary variable, based on which its [Formula: see text]-soliton solutions can be also directly derived. Furthermore, by considering multi-dimensional Riemann theta function, a lucid and straightforward generalization of the Hirota–Riemann method is presented to explicitly construct the multiperiodic wave solutions of the equation. Finally, the asymptotic properties of these periodic wave solutions are strictly analyzed to reveal the relationships between periodic wave solutions and soliton solutions.

Exact Solutions for a Coupled Korteweg–de Vries System

2016 ◽  
Vol 71 (11) ◽  
pp. 1053-1058
Author(s):  
Da-Wei Zuo ◽  
Hui-Xian Jia
Keyword(s):  
Exact Solutions ◽  
Shallow Water ◽  
Water Waves ◽  
Type Equation ◽  
Bilinear Forms ◽  
Interfacial Waves ◽  
Rational Solutions ◽  
Shallow Water Waves ◽  
De Vries

AbstractKorteweg–de Vries (KdV)-type equation can be used to characterise the dynamic behaviours of the shallow water waves and interfacial waves in the two-layer fluid with gradually varying depth. In this article, by virtue of the bilinear forms, rational solutions and three kind shapes (soliton-like, kink and bell, anti-bell, and bell shapes) for the Nth-order soliton-like solutions of a coupled KdV system are derived. Propagation and interaction of the solitons are analyzed: (1) Potential u shows three kind of shapes (soliton-like, kink, and anti-bell shapes); Potential v exhibits two type of shapes (soliton-like and bell shapes); (2) Interaction of the potentials u and v both display the fusion phenomena.

Extended F-Expansion Method for Solving the modified Korteweg-de Vries (mKdV) Equation

2020 ◽  
Vol 11 (1) ◽  
pp. 93-100
Author(s):  
Vina Apriliani ◽  
Ikhsan Maulidi ◽  
Budi Azhari
Keyword(s):  
Wave Equation ◽  
Exact Solutions ◽  
Internal Waves ◽  
Water Waves ◽  
Wave Equations ◽  
Elliptic Functions ◽  
Expansion Method ◽  
Mkdv Equation ◽  
Innovative Methods ◽  
De Vries

One of the phenomenon in marine science that is often encountered is the phenomenon of water waves. Waves that occur below the surface of seawater are called internal waves. One of the mathematical models that can represent solitary internal waves is the modified Korteweg-de Vries (mKdV) equation. Many methods can be used to construct the solution of the mKdV wave equation, one of which is the extended F-expansion method. The purpose of this study is to determine the solution of the mKdV wave equation using the extended F-expansion method. The result of solving the mKdV wave equation is the exact solutions. The exact solutions of the mKdV wave equation are expressed in the Jacobi elliptic functions, trigonometric functions, and hyperbolic functions. From this research, it is expected to be able to add insight and knowledge about the implementation of the innovative methods for solving wave equations. 

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