No abstract

As is well known, submerged horizontal cylinders can serve as waveguides for surface water waves. For large values of the wavenumberkin the direction of the cylinders, there is only one trapped wave. We construct asymptotics of these trapped modes and their frequencies ask→∞in the case of one or two submerged cylinders by means of reducing the initial problem to a system of integral equations on the boundaries and then solving them using a technique suggested by Zhevandrov and Merzon (2003).

We propose a simple method for constructing an asymptotic of an eigenvalue for the Klein-Gordon equation in the presence of a shallow potential well, reducing the initial problem to an integral equation and then by applying the method of Neumann series to solve it.

It is well known that gradient systems cannot have periodic orbits. In this work we find a general dynamical system on the plane without periodic orbits. We use Dulac's criterion that gives sufficient conditions for the non-existence of periodic orbits of dynamical systems in simply connected regions of the plane. Using a Dulac function we can rule out periodic orbits.

ABSTRACT. The purpose of this paper is to construct the asymptotic for natural frequencies of the Schrödinger equation using the method of Wentzel-Kramers-Brillouin (WKB).