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 $σ(T)$ 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 Schrödinger equations with odd pure-power nonlinearity. For both equations, we reach the lower bound $σ(T)\gtrsim T^{-\frac{1}{2}-ε}$, for any $ε>0$, thus improving all available results in the current literature.