Improved lower bound for the radius of analyticity for the modified KdV equation

Abstract

We investigate the initial value problem (IVP) associated to the modified Korteweg-de Vries equation (mKdV) in the defocusing scenario: \begin{equation*} \left\{\begin{array}{l} \partial_t u+ \partial_x^3u-u^2\partial_x(u) = 0, \quad x,t\in\mathbb{R}, \\ u(x,0) = u_0(x), \end{array}\right. \end{equation*} where $u$ is a real valued function and the initial data $u_0$ is analytic on $\mathbb{R}$ and has uniform radius of analyticity $\sigma_0$ in the spatial variable. It is well-known that the solution $u$ preserves its analyticity with the same radius $\sigma_0$ for at least some time span $0<T_0\le 1$. This local result was obtained in [Nonlinear Differ. Equ. Appl. (2024), 31--68] by proving a trilinear estimate in the Gevrey spaces $G^{\sigma, s}$, $s\geq \frac14$. Global in time behaviour of the solution and algebraic lower bound of the evolution of the radius of analyticity was also studied in authors' earlier works in [Nonlinear Differ. Equ. Appl. (2024), 31--68] and [J. Evol. Equ. 24 No. 42 (2024)] by constructing almost conserved quantities in the classical Gevrey space with $H^1$ and $H^2$ levels of Sobolev regularities. The present study aims to construct a new almost conservation law in the Gevrey space defined with a weight function $\cosh(\sigma|\xi|)$ and use it demonstrate that the local solution $u$ extends globally in time, and the radius of spatial analyticity is bounded from below by $c T^{-\frac{1}{2}}$, for any time $T\geq T_0$. The outcome of this paper represents an improvement on the one achieved by the authors' previous work in [J. Evol. Equ. 24 No. 42 (2024)].