On the radius of spatial analyticity for the modified Kawahara equation on the line

Abstract

First, by using linear and trilinear estimates in Bourgain type analytic and Gevrey spaces, the local well‐posedness of the Cauchy problem for the modified Kawahara equation on the line is established for analytic initial data u0false(xfalse) that can be extended as holomorphic functions in a strip around the x‐axis. Next we use this local result and a Gevrey approximate conservation law to prove that global solutions exist. Furthermore, we obtain explicit lower bounds for the radius of spatial analyticity r(t) given by rfalse(tfalse)≥ct−(4+δ), where δ>0 can be taken arbitrarily small and c is a positive constant.

Spatial Analyticity of Solutions to Korteweg–de Vries Type Equations

“…This article is a continuation of a number of previous works that were previously published in the same direction [15,16]. The main aim in the present paper is to treat the question of the well-posedness of (1), where w 0 (x) is analytic on the line and can be extended as holomorphic functions in a strip around the x−axis.…”

“…An effective method for studying lower bounds on the radius of analyticity, including this type of problem, was introduced in [14] for 1D Dirac-Klein-Gordon equations. It was applied in [15] to the modified Kawahara equation and in [16] to the non-periodic KdV equation. (For more details, please see [17][18][19][20].…”

“…The proofs of the next lemmas, for θ = 0, are developed in [16], for θ > 0, using operator A in (15).…”

Well-posedness and regularity of the fifth order Kadomtsev–Petviashvili I equation in the analytic Bourgain spaces

“…The method used here for proving lower bounds on the radius of analyticity was introduced in [8] in the study of the 1D Dirac-Klein-Gordon equations. It was applied to the modified Kawahara equation [5] and the non-periodic KdV equation in [7], to the dispersion-generalized periodic KdV equation in [2], the Ostrovsky equation [1], and to the quartic generalized KdV equation on the line in [9]. The rest of the paper is organized as follows: in section 2, we introduce the various tools which will be used in the proofs of our main theorems.…”

On the radius of spatial analyticity for defocusing nonlinear Schrödinger equations

“…More precisely, if the initial data are real-analytic and have a uniform radius of analyticity σ 0 > 0, so there is a holomorphic extension of the data to a complex strip S σ0 = {x + iy : x, y ∈ R d , |y 1 |, |y 2 |, • • • , |y d | < σ 0 }, then we may ask whether or not and up to what degree the solution at some later time t preserves the initial analyticity; we would like to estimate the radius of analyticity of the solution at time t, σ(t), which is possibly shrinking. This type of question was first introduced by Kato and Masuda [16] in 1986 and there are plenty of works for nonlinear dispersive equations such as the KP equation [3], KdV type equations [4,5,24,28,14,22,2], Schrödinger equations [6,27,1], and Klein-Gordon equations [18].…”