Spatial Analyticity of Solutions to Korteweg–de Vries Type Equations

Keltoum Bouhali,Khdija O Taha,Yousif Altayeb,Khadiga W Tajer,Abdelkader Moumen

doi:10.3390/mca26040075

Abstract

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.

Highlights

Introduction and Main ResultsThe study of Korteweg–de Vries (KdV) type equations in Bourgain type spaces is an important task, both from a theoretical point of view and from the point of view of applications
We introduce the analytic Gevrey–Bourgain spaces associated with the KdV equation
The proof of Theorem 2 is completed

Summary

Introduction and Main Results

The study of Korteweg–de Vries (KdV) type equations in Bourgain type spaces is an important task, both from a theoretical point of view (existence and uniqueness of solution theorems) and from the point of view of applications. In view of the Paley–Wiener theorem, it is natural to take initial data in Gθ,s and obtain a better understanding of the behavior of solutions as we try to extend it globally in time This means that, given w0 ∈ Gθ,s for some initial radius θ > 0, we want to estimate the behavior of the radius of analyticity θ(T) over time. Assume that w0 ∈ Gθ0,s; the solutions in Theorem 1 can be extended to be global in time, and, for any T > 0, we have w ∈ C [−T , T ], Gθ(T ),s with θ(T ) = min θ0, C1T (6−σ0) , where σ0 > 0 can be taken as arbitrarily small and C1 > 0 is a constant depending on w0, θ0, s and σ0.

Linear Estimates

Existence of Solution

Continuous Dependence on the Initial Data

Approximate Conservation Law

Proof of Theorem 2

The General Case