Decay of the radius of spatial analyticity for the modified KdV equation and the nonlinear Schrödinger equation with third order dispersion (2307.09096v2)

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+\mu u^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 $\mu=\pm 1$, and the cubic nonlinear Schr\"odinger equation with third order dispersion (tNLS equation in short) \begin{equation*} \label{t-NLS} \left{\begin{array}{l} \partial_t v+i\alpha \partial_x^2v+\beta \partial_x^3v+i\gamma |v|^2v = 0, \quad x\in\mathbb{R},\; t\in\mathbb{R} , \ v(x,0) = v_0(x), \end{array}\right. \end{equation*} where $\alpha, \beta$ and $\gamma$ 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 $\sigma_0$ 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 $\sigma_0$ till some lifespan $0<T_0\le 1$. We also consider the evolution of the radius of spatial analyticity $\sigma(t)$ when the local solution extends globally in time and prove that for any time $T\ge T_0$ it is bounded from below by $c T^{-\frac43}$, for the mKdV equation in the defocusing case ($\mu = -1$) and by $c T^{-(4+\varepsilon)}$, $\varepsilon\>0$, for the tNLS equation. The result for the mKdV equation improves the one obtained in [ J. L. Bona, Z. Gruji\'c and H. Kalisch, Algebraic lower bounds for the uniform radius of spatial analyticity for the generalized KdV equation, Ann Inst. H. Poincar\'e 22 (2005) 783--797] and, as far as we know, the result for the tNLS equation is the new one.