We prove existence results for travelling waves in discrete, damped, dc-driven sine-Gordon equations with periodic boundary conditions.

Over twenty years ago, Marc Kac posed what is arguably one of the simplest inverse problems in pure mathematics: "Can one hear the shape of a drum?" [19]. Mathematically, the question is formulated as follows. Let /2 be a simply connected, plane domain (the drumhead) bounded by a smooth curve 7, and consider the wave equation on /2 with Dirichlet boundary condition on 7 (the drumhead is clamped...

In this paper we consider the one-parameter family of Bona-Smith systems, which belongs to the class of Boussinesq systems modelling two-way propagation of long waves of small amplitude on the surface of water in a channel. We study three initial-boundary-value problems for these systems, corresponding, respectively, to nonhomogeneous Dirichlet, reflection, and periodic boundary conditions pose...

Diophantine criteria occur naturally in the theory of partial differential equations through the notorious problem of small denominators. An extensive treatment of such problems in the theory of PDEs can be found, e.g., in [6]. In this paper, we are interested in a Diophantine problem related to an inhomogeneous wave equation in n spatial and one temporal dimension with periodic boundary condit...

The Korteweg-de Vries (KdV) equation with periodic boundary conditions is considered. The interaction of a periodic solitary wave (cnoidal wave) with high frequency radiation of finite energy (L-norm) is studied. It is proved that the interaction of low frequency component (cnoidal wave) and high frequency radiation is weak for finite time in the following sense: the radiation approximately sat...

In [2], two of the authors and W. Meeks found examples of translation-invariant, embedded minimal surfaces with an infinite number of topological ends. For each k > 0, a surfaceMk was constructed, which was invariant with respect to a translation parallel to the x3-axis, and under a rotation group of order k+ 1 around the x3-axis. The method of construction was generalized in [3] to obtain the ...

We show how to stabilize the uniform oscillations of the complex Ginzburg–Landau equation with periodic boundary conditions by means of some global delayed feedback. The proof is based on an abstract pseudo-linearization principle and a careful study of the spectrum of the linearized operator.

The significance of 2π is that it is a cluster point of the length spectrum from the left (t < 2π), as described in more detail below. In brief, the length spectrum is the closure of the set of lengths of periodic geodesics. The number 2π represents the length of the (generalized) geodesic that follows the circumference of the circle. The number 8 is the geodesic that follows a diameter of the ...