Radiation solution of the modified Korteweg–de Vries equation

1985 ◽  
Vol 63 (4) ◽  
pp. 532-544
Author(s):  
R. H. Enns ◽  
S. S. Rangnekar
Keyword(s):  
Integral Equation ◽  
Temporal Evolution ◽  
Vries Equation ◽  
Wave Interaction ◽  
Third Order ◽  
Exact Agreement ◽  
De Vries ◽  
Scattering Study ◽  
Interaction Problem ◽  
Input Potential

An inverse scattering study of the radiation solution of the modified Korteweg–de Vries equation is carried out for a simple illustrative example. Specifically, extending the expansion approach (the reflection coefficient being expanded in powers of the area of the input potential) that we pioneered on the 3-wave interaction problem and recently applied to the study of sine-Gordon and sinh-Gordon dynamics, we obtain the complete spatial and temporal evolution of the modified KdV solution up to third order in the expansion. The solution and, in particular, its asymptotic (t → ∞) behaviour are discussed and a comparison is made with the asymptotic analysis of Ablowitz and Newell. The nonintegral contributions to the radiation solution are found to be in exact agreement as t → ∞ with Ablowitz and Newell's steepest descents approximation (evaluated for the same model) for the kernel of the Marchenko integral equation.

Abundant rogue wave solutions for the (2 + 1)-dimensional generalized Korteweg–de Vries equation

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Huanhuan Lu ◽  
Yufeng Zhang
Keyword(s):  
Vries Equation ◽  
Rogue Wave ◽  
Rogue Waves ◽  
Third Order ◽  
First Order ◽  
Wave Solutions ◽  
De Vries ◽  
Rogue Wave Solutions ◽  
Bilinear Formulation ◽  
Hirota Bilinear

AbstractIn this paper, we analyse two types of rogue wave solutions generated from two improved ansatzs, to the (2 + 1)-dimensional generalized Korteweg–de Vries equation. With symbolic computation, the first-order rogue waves, second-order rogue waves, third-order rogue waves are generated directly from the first ansatz. Based on the Hirota bilinear formulation, another type of one-rogue waves and two-rogue waves can be obtained from the second ansatz. In addition, the dynamic behaviours of obtained rogue wave solutions are illustrated graphically.

scholarly journals New Cubic B-spline Approximations for Solving Non-linear Third-order Korteweg-de Vries Equation

2019 ◽  
Vol 12 (6) ◽  
pp. 1-9
Author(s):  
Muhammad Abbas ◽  
Muhammad Kashif Iqbal ◽  
Bushra Zafar ◽  
Shazalina Binti Mat Zin ◽  
◽  
...  
Keyword(s):  
Vries Equation ◽  
Third Order ◽  
B Spline ◽  
Non Linear ◽  
De Vries

scholarly journals Investigation of Interaction Solutions for Modified Korteweg-de Vries Equation by Consistent Riccati Expansion Method

2019 ◽  
Vol 2019 ◽  
pp. 1-8
Author(s):  
Jin-Fu Liang ◽  
Xun Wang
Keyword(s):  
Vries Equation ◽  
Soliton Solutions ◽  
Wave Interaction ◽  
Expansion Method ◽  
Periodic Wave ◽  
Mkdv Equation ◽  
Jacobi Elliptic Function ◽  
Interaction Solutions ◽  
De Vries ◽  
Consistent Riccati Expansion

A consistent Riccati expansion (CRE) method is proposed for obtaining interaction solutions to the modified Korteweg-de Vries (mKdV) equation. Using the CRE method, it is shown that interaction solutions such as the soliton-tangent (or soliton-cotangent) wave cannot be constructed for the mKdV equation. More importantly, exact soliton-cnoidal periodic wave interaction solutions are presented. While soliton-cnoidal interaction solutions were found to degenerate to special resonant soliton solutions for the values of modulus (n) closer to one (upper bound of modulus) in the Jacobi elliptic function, a normal kink-shaped soliton was observed for values of n closer to zero (lower bound).

scholarly journals Spatial Analyticity of Solutions to Korteweg–de Vries Type Equations

2021 ◽  
Vol 26 (4) ◽  
pp. 75
Author(s):  
Keltoum Bouhali ◽  
Abdelkader Moumen ◽  
Khadiga W. Tajer ◽  
Khdija O. Taha ◽  
Yousif Altayeb
Keyword(s):  
Differential Equation ◽  
Nonlinear Waves ◽  
Vries Equation ◽  
Nonlinear Partial Differential Equation ◽  
Holomorphic Functions ◽  
Well Posedness ◽  
Initial Value ◽  
Third Order ◽  
De Vries ◽  
Type Spaces

The Korteweg–de Vries equation (KdV) is a mathematical model of waves on shallow water surfaces. It is given as third-order nonlinear partial differential equation and plays a very important role in the theory of nonlinear waves. It was obtained by Boussinesq in 1877, and a detailed analysis was performed by Korteweg and de Vries in 1895. In this article, by using multi-linear estimates in Bourgain type spaces, we prove the local well-posedness of the initial value problem associated with the Korteweg–de Vries equations. The solution is established online for analytic initial data w0 that can be extended as holomorphic functions in a strip around the x-axis. A procedure for constructing a global solution is proposed, which improves upon earlier results.

scholarly journals On the existence of the resolvent and separability of a class of the Korteweg-de Vriese type linear singular operators

2021 ◽  
Vol 101 (1) ◽  
pp. 87-97
Author(s):  
М.B. Muratbekov ◽  
◽  
A.O. Suleimbekova ◽  
Keyword(s):  
Vries Equation ◽  
Mathematical Physics ◽  
Third Order ◽  
Aeration Zone ◽  
Singular Operators ◽  
Main Equation ◽  
Protein Molecules ◽  
De Vries ◽  
Hydrolysis Of ◽  
Moisture Potential

Partial differential equations of the third order are the basis of mathematical models of many phenomena and processes, such as the phenomenon of energy transfer of hydrolysis of adenosine triphosphate molecules along protein molecules in the form of solitary waves, i.e. solitons, the process of transferring soil moisture in the aeration zone, taking into account its movement against the moisture potential. In particular, this class includes the nonlinear Korteweg-de Vries equation, which is the main equation of modern mathematical physics. It is known that various problems have been studied for the Korteweg-de Vries equation and many fundamental results obtained. In this paper, issues about the existence of a resolvent and separability (maximum smoothness of solutions) of a class of linear singular operators of the Korteweg-de Vries type in the case of an unbounded domain with strongly increasing coefficients are investigated.

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