scholarly journals Local well-posedness for the periodic Korteweg-de Vries equation in analytic Gevrey classes

2012 ◽  
Vol 11 (3) ◽  
pp. 1097-1109 ◽  
Author(s):  
Qifan Li ◽  
Keyword(s):  
Vries Equation ◽  
Well Posedness ◽  
Gevrey Classes ◽  
De Vries

scholarly journals On unconditional well-posedness for the periodic modified Korteweg–de Vries equation

2019 ◽  
Vol 71 (1) ◽  
pp. 147-201 ◽  
Author(s):  
Luc MOLINET ◽  
Didier PILOD ◽  
Stéphane VENTO
Keyword(s):  
Vries Equation ◽  
Well Posedness ◽  
De Vries

scholarly journals Self-Similar Dynamics for the Modified Korteweg–de Vries Equation

2020 ◽  
Author(s):  
Simão Correia ◽  
Raphaël Côte ◽  
Luis Vega
Keyword(s):  
Vries Equation ◽  
Well Posedness ◽  
Critical Space ◽  
Asymptotic Description ◽  
De Vries ◽  
Self Similar ◽  
Similar Solutions

Abstract We prove a local well-posedness result for the modified Korteweg–de Vries equation in a critical space designed so that is contains self-similar solutions. As a consequence, we can study the flow of this equation around self-similar solutions: in particular, we give an asymptotic description of small solutions as $t \to +\infty$.

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.

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