Analytic well-posedness of periodic gKdV

Abstract

In the periodic case, it is proved that the Cauchy problem for the generalized Korteweg-de Vries equation (gKdV) is locally well-posed in a class of analytic functions that can be extended holomorphically in a symmetric strip of the complex plane around the x-axis. Thus, the uniform analyticity radius of the solution does not change as time progresses. Also, information about the regularity of the solution in the time variable is provided.

On the radius of spatial analyticity for cubic nonlinear Schrödinger equations

“…For large times on the other hand we use the idea introduced in [17] (see also [16]) to show that σ(t ) can decay no faster than 1/|t | as |t | → ∞. For studies on related issues for nonlinear partial differential equations see for instance [2,3,10,11,12,13,15].…”

Lower Bounds on the Radius of Spatial Analyticity for the KdV Equation

“…This question has received some attention in the case of the KdV equation and its generalizations. For short times, it is known that the radius of analyticity remains at least as large as the initial radius; see Grujić and Kalisch [6] for the non-periodic case, and also Li [15], Himonas and Petronilho [8], and Hannah, Himonas and Petronilho [7] for the periodic case. For the global problem, the non-periodic case was studied by Bona, Grujić and Kalisch in [2], where it was shown that the radius of analyticity for the KdV equation can decay no faster than t −12 as t → ∞ (see Theorem 4 and Corollary 2 in [2]).…”

On the radius of spatial analyticity for semilinear symmetric hyperbolic systems

“…Motivation for this type of study comes from the global analytic theory of nonlinear evolution PDE started with the paper by Kato and Masuda [28] and based on the proof of the existence of global analytic solutions in the space variables for the related Cauchy problem. After [28], many authors proved results of this type for several models as the (generalized) Korteweg-de Vries equation, the Euler equations, the Benjamin-Ono equation, the nonlinear Schrödinger equation, see for instance [5,8,20,21,22,23,24,25,26]. A parallel study in an elliptic setting devoted in particular to the analyticity of travelling waves has been developed in [6,10,11,13,14].…”