Their topics include solitary wave generation due to passage through the local resonance,

asymptotic expansions of singular perturbation theory, the Kelvin-Helmholtz instability, and nonlinear Cauchy problems for elliptic equations.

By the

asymptotic expansions above we find that [[rho].sub.OO0]([k.sub.min], L, [eta],p, a, b) > [[rho].sub.OO0]([k.sub.min], L, [eta],p*, a, b) if [C.sub.p] > [C*.sub.p] and [[rho].sub.OO0]([bar.k](L, [eta],p),L,[eta],p, a, b) > [[rho].sub.OO0]([bar.k](L, [eta],p*), L, [eta],p*, a, b,) if [C.sub.p] < [C*.sub.p], which concludes the proof of the asymptotic optimality.

Although the Lindstedt-Poincare method gives uniformly valid

asymptotic expansions for periodic solutions of weakly nonlinear oscillations, i.e., 0 < [epsilon] < [C.sub.1], the technique does not work if the amplitude of the oscillation is a function of time (see [16, 17]).

Next lemma unveils the

asymptotic expansions of [mathematical expression not reproducible], where

Erdelyi,

Asymptotic Expansions, Dover, New York, NY, USA, 1956.

First we give the

asymptotic expansions of [mathematical expression not reproducible] for [xi] [right arrow] 0.

Various methods are employed to solve singular perturbation problems analytically, numerically, or asymptotically such as the method of matched

asymptotic expansions (MMAE), the method of multiple scales, the method of WKB approximation, Poincare-Lindstedt method and periodic averaging method.

Third, the ideals in establishment of

asymptotic expansions for each problem are different.

The search for

asymptotic expansions and approximations of special functions is a very classical vein of research and is of great relevance in pure mathematics, in numerical analysis, mathematical physics, and the applied sciences (see, for instance, of course with no pretence of completion [1-4]).

So, guess what does it take to calculate these numbers and their corresponding

asymptotic expansions?

Our current effort in this paper is to further explore the limiting behavior of the process [x.sup.[epsilon], [delta]](x) by constructing

asymptotic expansions of its probability densities, which are associated with the adjoint operator [L.sup.*].