Asymptotic approximation of a highly oscillatory integral with application to the canonical catastrophe integrals

We consider the highly oscillatory integralF ( w ) : = ∫ − ∞ ∞ e i w ( t K + 2 + e i θt p ) g ( t ) d t $F(w):=\int _{-\infty }^\infty e^{iw(t^{K+2}+e^{i\theta }t^p)}g(t)dt$ for large positive values of w ,− π < θ ≤ π $-\pi <\theta \le \pi$ , K and p positive integers with1 ≤ p ≤ K $1\le p\le K$ , andg ( t ) $g(t)$ an entire function. The standard saddle point method is complicated and we use here a simplified version of this method introduced by López et al. We derive an asymptotic approximation of this integral whenw → + ∞ $w\rightarrow +\infty$ for general values of K and p in terms of elementary functions, and determine the Stokes lines. Forp ≠ 1 $p\ne 1$ , the asymptotic behavior of this integral may be classified in four different regions according to the even/odd character of the couple of parameters K and p ; the special casep = 1 $p=1$ requires a separate analysis. As an important application, we consider the family of canonical catastrophe integralsΨ K ( x 1 , x 2 , … , x K ) $\Psi _K(x_1,x_2,\ldots ,x_K)$ for large values of one of its variables, sayx p $x_p$ , and bounded values of the remaining ones. This family of integrals may be written in the formF ( w ) $F(w)$ for appropriate values of the parameters w , θ and the functiong ( t ) $g(t)$ . Then, we derive an asymptotic approximation of the family of canonical catastrophe integrals for large| x p | $\vert x_p\vert$ . The approximations are accompanied by several numerical experiments. The asymptotic formulas presented here fill up a gap in the NIST Handbook of Mathematical Functions by Olver et al.