Exponential asymptotic expansions for nonlinear differential equations

An exponential method for obtaining asymptotic expansions of the solution of a linear differential equation was presented by Brull and Soler in [3], We extend this method to the nonlinear differential equation of the type $L\left ( u \right ) + {\Sigma _1}^Nf\left ( {x, t} \right ){u^n} = 0$, where $t$ is the small parameter. Three examples are used to illustrate the technique and to explain how uniformly valid expansions may be obtained.