  In several cases of nonlinear dispersive PDEs, the difference between the
nonlinear and linear evolutions with the same initial data, i.e. the integral
term in Duhamel's formula, exhibits improved regularity. This property is
usually called nonlinear smoothing. We employ the method of infinite iterations
of normal form reductions to obtain a very general theorem yielding existence
of nonlinear smoothing for dispersive PDEs, contingent only on establishing two
particular bounds. We then apply this theorem to show that the nonlinear
smoothing property holds, depending on the regularity of the initial data, for
five classical dispersive equations: in $\mathbb{R}$, the cubic nonlinear
Schr\"odinger, the Korteweg-de Vries, the modified Korteweg-de Vries and the
derivative Schr\"odinger equations; in $\mathbb{R}^2$, the modified
Zakharov-Kuznetsov equation. For the aforementioned one-dimensional equations,
this unifying methodology matches or improves the existing nonlinear smoothing
results, while enlarging the ranges of Sobolev regularities where the property
holds.
