  It is well known that Lawson methods suffer from a severe order reduction
when integrating initial boundary value problems where the solutions are not
periodic in space or do not satisfy enough conditions of annihilation on the
boundary. However, in a previous paper, a modification of Lawson quadrature
rules has been suggested so that no order reduction turns up when integrating
linear problems subject to even time-dependent boundary conditions. In this
paper, we describe and thoroughly analyse a technique to avoid also order
reduction when integrating nonlinear problems. This is very useful because,
given any Runge-Kutta method of any classical order, a Lawson method can be
constructed associated to it for which the order is conserved.
