Decay of the radius of spatial analyticity for the modified KdV equation and the nonlinear Schrödinger equation with third order dispersion

Abstract

We consider the initial value problems (IVPs) for the modified Korteweg-de Vries (mKdV) equation \begin{equation*} \label{mKdV} \left\{\begin{array}{l} \partial_t u+ \partial_x^3u+μu^2\partial_xu =0, \quad x\in\mathbb{R},\; t\in \mathbb{R} , \\ u(x,0) = u_0(x), \end{array}\right. \end{equation*} where $u$ is a real valued function and $μ=\pm 1$, and the cubic nonlinear Schrödinger equation with third order dispersion (tNLS equation in short) \begin{equation*} \label{t-NLS} \left\{\begin{array}{l} \partial_t v+iα\partial_x^2v+β\partial_x^3v+iγ|v|^2v = 0, \quad x\in\mathbb{R},\; t\in\mathbb{R} , \\ v(x,0) = v_0(x), \end{array}\right. \end{equation*} where $α, β$ and $γ$ are real constants and $v$ is a complex valued function. In both problems, the initial data $u_0$ and $v_0$ are analytic on $\mathbb{R}$ and have uniform radius of analyticity $σ_0$ in the space variable. We prove that the both IVPs are locally well-posed for such data by establishing an analytic version of the trilinear estimates, and showed that the radius of spatial analyticity of the solution remains the same $σ_0$ till some lifespan $0 0$, for the tNLS equation. The result for the mKdV equation improves the one obtained in [ J. L. Bona, Z. Grujić and H. Kalisch, Algebraic lower bounds for the uniform radius of spatial analyticity for the generalized KdV equation, Ann Inst. H. Poincaré 22 (2005) 783--797] and, as far as we know, the result for the tNLS equation is the new one.