Boundary Stabilization of the Korteweg-de Vries Equation and the Korteweg-de Vries-Burgers Equation

2012 ◽  
Vol 118 (1) ◽  
pp. 25-47 ◽  
Author(s):  
Chaohua Jia ◽  
Bing-Yu Zhang
Keyword(s):  
Burgers Equation ◽  
Vries Equation ◽  
Boundary Stabilization ◽  
De Vries

scholarly journals Modelling and nonlinear boundary stabilization of the modified generalized Korteweg–de Vries–Burgers equation

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
N. Smaoui ◽  
B. Chentouf ◽  
A. Alalabi
Keyword(s):  
Nonlinear Boundary ◽  
Burgers Equation ◽  
Stabilization Problem ◽  
Boundary Stabilization ◽  
Wave Approximation ◽  
Long Wave ◽  
Control Schemes ◽  
De Vries ◽  
Long Wave Approximation ◽  
Stability Of The Solution

Abstract In this paper, we study the modelling and nonlinear boundary stabilization problem of the modified generalized Korteweg–de Vries–Burgers equation (MGKdVB) when the spatial domain is $[0,1]$ [ 0 , 1 ] . First, the MGKdVB equation is derived using the long-wave approximation and perturbation method. Then, two nonlinear boundary controllers are proposed for this equation and the $L^{2} $ L 2 -global exponential stability of the solution is shown. Numerical simulations are given to illustrate the efficiency of the developed control schemes.

Shock wave occurrence and soliton propagation in polariton condensates

2017 ◽  
Vol 95 (12) ◽  
pp. 1234-1238 ◽  
Author(s):  
A.M. Belounis ◽  
S. Kessal
Keyword(s):  
Shock Wave ◽  
Burgers Equation ◽  
Vries Equation ◽  
Reductive Perturbation Method ◽  
Pitaevskii Equation ◽  
Soliton Propagation ◽  
Modified Burgers Equation ◽  
Reductive Perturbation ◽  
De Vries ◽  
Dispersionless Limit

We study the effects of the gain and the loss of polaritons on the wave propagation in polariton condensates. This system is described by a modified Gross–Pitaevskii equation. In the case of small damping, we use the reductive perturbation method to transform this equation; we get a modified Burgers equation in the dispersionless limit and a damped Korteweg – de Vries equation in a more general case. We demonstrate that the shock wave occurrence depends on the gain and the loss of polaritons in the dispersionless polariton condensate. The resolution of the damped Korteweg – de Vries equation shows that the soliton behaves like a damped wave in the case of a constant damping. Based on an asymptotic solution, the survival time and the distance traveled by this soliton are evaluated. We solve the damped Korteweg – de Vries equation and the modified Gross–Pitaevskii numerically to validate the analytical calculations and discuss especially the soliton propagation in the system.

Integration of the nonlinear modified Korteweg-de Vries equation with a loaded term

2020 ◽  
Vol 2020 (2) ◽  
pp. 85-98
Author(s):  
A.B. Khasanov ◽  
T.J. Allanazarova
Keyword(s):  
Vries Equation ◽  
De Vries ◽  
Loaded Term

Convergence of the Rosenau-Korteweg-de Vries Equation to the Korteweg-de Vries One

2020 ◽  
Author(s):  
Giuseppe Maria Coclite ◽  
Lorenzo di Ruvo
Keyword(s):  
A Priori Estimates ◽  
Vries Equation ◽  
A Priori ◽  
Diffusion Parameter ◽  
Priori Estimates ◽  
De Vries ◽  
Wall Interactions

The Rosenau-Korteweg-de Vries equation describes the wave-wave and wave-wall interactions. In this paper, we prove that, as the diffusion parameter is near zero, it coincides with the Korteweg-de Vries equation. The proof relies on deriving suitable a priori estimates together with an application of the Aubin-Lions Lemma.

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