Stability of the Korteweg-de Vries solitary wave solution

1994 ◽  
Vol 15 (9) ◽  
pp. 885-886
Author(s):  
Lü Xian-qing
Keyword(s):  
Solitary Wave ◽  
Wave Solution ◽  
Solitary Wave Solution ◽  
De Vries

scholarly journals Formation of wave packets in the Ostrovsky equation for both normal and anomalous dispersion

2016 ◽  
Vol 472 (2185) ◽  
pp. 20150416 ◽  
Author(s):  
Roger Grimshaw ◽  
Yury Stepanyants ◽  
Azwani Alias
Keyword(s):  
Solitary Wave ◽  
Wave Solution ◽  
Solitary Wave Solution ◽  
Anomalous Dispersion ◽  
Essential Difference ◽  
De Vries ◽  
Long Time ◽  
Ostrovsky Equation ◽  
Phase And Group Velocities ◽  
Group Velocities

It is well known that the Ostrovsky equation with normal dispersion does not support steady solitary waves. An initial Korteweg–de Vries solitary wave decays adiabatically through the radiation of long waves and is eventually replaced by an envelope solitary wave whose carrier wave and envelope move with different velocities (phase and group velocities correspondingly). Here, we examine the same initial condition for the Ostrovsky equation with anomalous dispersion, when the wave frequency increases with wavenumber in the limit of very short waves. The essential difference is that now there exists a steady solitary wave solution (Ostrovsky soliton), which in the small-amplitude limit can be described asymptotically through the solitary wave solution of a nonlinear Schrödinger equation, based at that wavenumber where the phase and group velocities coincide. Long-time numerical simulations show that the emergence of this steady envelope solitary wave is a very robust feature. The initial Korteweg–de Vries solitary wave transforms rapidly to this envelope solitary wave in a seemingly non-adiabatic manner. The amplitude of the Ostrovsky soliton strongly correlates with the initial Korteweg–de Vries solitary wave.

Periodic Wave Solutions of a Generalized KdV-mKdV Equation with Higher-Order Nonlinear Terms

2010 ◽  
Vol 65 (8-9) ◽  
pp. 649-657 ◽  
Author(s):  
Zi-Liang Li
Keyword(s):  
Solitary Wave ◽  
Wave Solution ◽  
Solitary Wave Solution ◽  
Periodic Wave ◽  
Higher Order ◽  
Periodic Wave Solution ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
De Vries ◽  
Ode Method

The Jacobin doubly periodic wave solution, the Weierstrass elliptic function solution, the bell-type solitary wave solution, the kink-type solitary wave solution, the algebraic solitary wave solution, and the triangular solution of a generalized Korteweg-de Vries-modified Korteweg-de Vries equation (GKdV-mKdV) with higher-order nonlinear terms are obtained by a generalized subsidiary ordinary differential equation method (Gsub-ODE method for short). The key ideas of the Gsub-ODE method are that the periodic wave solutions of a complicated nonlinear wave equation can be constructed by means of the solutions of some simple and solvable ODE which are called Gsub-ODE with higherorder nonlinear terms

Analytical Solitary Wave Solution of the Dust Ion Acoustic Waves for the Damped Forced Korteweg–de Vries Equation in Superthermal Plasmas

2018 ◽  
Vol 73 (2) ◽  
pp. 151-159 ◽  
Author(s):  
Prasanta Chatterjee ◽  
Rustam Ali ◽  
Asit Saha
Keyword(s):  
Solitary Wave ◽  
Acoustic Waves ◽  
Wave Solution ◽  
Solitary Wave Solution ◽  
Perturbation Technique ◽  
Dusty Plasmas ◽  
Periodic Force ◽  
Ion Acoustic ◽  
De Vries ◽  
Dia Waves

AbstractAnalytical solitary wave solution of the dust ion acoustic (DIA) waves was studied in the framework of the damped forced Korteweg–de Vries (DFKdV) equation in superthermal collisional dusty plasmas. The reductive perturbation technique was applied to derive the DKdV equation. It is observed that both the rarefactive and compressive solitary wave solutions are possible for this plasma model. The effects of κ and the strength (f0) and frequency (ω) of the external periodic force were studied on the analytical solitary wave solution of the DIA waves. It is observed that the parameters κ, f0 and ω have significant effects on the structure of the damped forced DIA solitary waves. The results of this study may have relevance in laboratory plasmas as well as in space plasmas.

scholarly journals Abundant Explicit and Exact Solutions for the Variable Coefficient mKdV Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Xiaoxiao Zheng ◽  
Yadong Shang ◽  
Yong Huang
Keyword(s):  
Solitary Wave ◽  
Elliptic Function ◽  
Constant Coefficient ◽  
Wave Solution ◽  
Integral Method ◽  
Solitary Wave Solution ◽  
First Integral ◽  
Mkdv Equation ◽  
Variable Coefficients ◽  
First Integral Method

This paper is concerned with the variable coefficients mKdV (VC-mKdV) equation. First, through some transformation we convert VC-mKdV equation into the constant coefficient mKdV equation. Then, using the first integral method we obtain the exact solutions of VC-mKdV equation, such as rational function solutions, periodic wave solutions of triangle function, bell-shape solitary wave solution, kink-shape solitary wave solution, Jacobi elliptic function solutions, and Weierstrass elliptic function solution. Furthermore, with the aid of Mathematica, the extended hyperbolic functions method is used to establish abundant exact explicit solution of VC-mKdV equation. By the results of the equation, the first integral method and the extended hyperbolic function method are extended from the constant coefficient nonlinear evolution equations to the variable coefficients nonlinear partial differential equation.

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