Nonlinear smoothing implies improved lower bounds on the radius of spatial analyticity for nonlinear dispersive equations

We provide a roadmap to establish improved lower bounds on the decay rate of the uniform radius of analyticity $\sigma(T)$ for a given nonlinear dispersive equation, reducing the problem to the derivation of nonlinear smoothing estimates with a specific distribution of extra derivatives. We apply this strategy for both the defocusing generalized KdV and the nonlinear Schr\"odinger equations with odd pure-power nonlinearity. For both equations, we reach the lower bound $\sigma(T)\gtrsim T^{-\frac{1}{2}-\epsilon}$, for any $\epsilon>0$, thus improving all available results in the current literature.