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

$$\begin{aligned} \left\{ \begin{array}{l} \partial _t u+ \partial _x^3u+\mu u^2\partial _xu =0, \quad x\in \mathbb {R},\; t\in \mathbb {R}, \\ u(x,0) = u_0(x), \end{array}\right. \end{aligned}$$

where u is a real valued function and \(\mu =\pm 1\), and the cubic nonlinear Schrödinger equation with third order dispersion (tNLS equation in short)

$$\begin{aligned} \left\{ \begin{array}{l} \partial _t v+i\alpha \partial _x^2v+\beta \partial _x^3v+i\gamma |v|^2v = 0, \quad x\in \mathbb {R},\; t\in \mathbb {R}, \\ v(x,0) = v_0(x), \end{array}\right. \end{aligned}$$

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 \(c T^{-\frac{4}{3}}\), 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 Bona et al. (Ann Inst Henri Poincaré 22:783–797, 2005) and, as far as we know, the result for the tNLS equation is the new one.