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

We investigate the initial value problem (IVP) associated to the modified Korteweg-de Vries equation (mKdV) in the defocusing scenario: {equation*} \{{array}{l} ∂_t u+ ∂_x^3u-u^2∂_x(u) = 0, x,t, \\ u(x,0) = u_0(x), {array}. {equation*} where u is a real valued function and the initial data u₀ is analytic on R and has uniform radius of analyticity σ₀ in the spatial variable. It is well-known that the solution u preserves its analyticity with the same radius σ₀ for at least some time span 0<T₀≤ 1. This local result was obtained in [Nonlinear Differ. Equ. Appl. (2024), 31--68] by proving a trilinear estimate in the Gevrey spaces Gσ, s, s≥ 14. 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 H¹ and H² levels of Sobolev regularities. The present study aims to construct a new almost conservation law in the Gevrey space defined with a weight function (σ|ξ|) 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-1/2, for any time T≥ T₀. 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)].