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We consider the initial value problems (IVPs) for the modified Korteweg-deVries (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 nonlinearSchr\"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 valuedfunction. 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 spacevariable.  We prove that the both IVPs are locally well-posed for such data byestablishing an analytic version of the trilinear estimates, and showed thatthe 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 radiusof spatial analyticity $\sigma(t)$ when the local solution extends globally intime and prove that for any time $T\ge T_0$ it is bounded from below by $cT^{-\frac43}$, for the mKdV equation in the defocusing case ($\mu = -1$) and by$c T^{-(4+\varepsilon)}$, $\varepsilon>0$, for the tNLS equation. The resultfor the mKdV equation improves the one obtained in [ J. L. Bona, Z. Gruji\'cand H. Kalisch, Algebraic lower bounds for the uniform radius of spatialanalyticity 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 newone.

Decay of the radius of spatial analyticity for the modified KdV equation and the nonlinear Schr\"odinger equation with third order dispersion