On the Asymptotics of a Bessel-Type Integral Having Applications in Wave Run-Up Theory

Rapidly oscillating integrals of the form
\begin{equation*} I(r,h)=\frac{1}{2\pi}\int_{-\pi}^{\pi} e^{\tfrac ih F(r\cos\phi)} G(r\cos\phi) \,d\phi, \end{equation*}
where $F(r)$ is a real-valued function with nonvanishing derivative, arise when constructing asymptotic solutions of problems with nonstandard characteristics such as the Cauchy problem with spatially localized initial data for the wave equation with velocity degenerating on the boundary of the domain; this problem describes the run-up of tsunami waves on a shallow beach in the linear approximation. The computation of the asymptotics of this integral as $h\to0$ encounters difficulties owing to the fact that the stationary points of the phase function $F(r\cos\phi)$ become degenerate for $r=0$. For this integral, we construct an asymptotics uniform with respect to $r$ in terms of the Bessel functions $\mathbf{J}_0(z)$ and $\mathbf{J}_1(z)$ of the first kind.