Simplest breather

The simplest Klein-Gordon (KG) breather is a compacton on a string subject to the force of gravity in a frictionless V-shaped trough. Its dynamics, spectrum, and energy are discussed and it is compared to sine-Gordon breathers. A generalization of this problem consists of a charged string subject to the electrostatic force of two semi-infinite coplanar charged planes separated by a gap of constant width. For motion in the midplane between these planes, the string's displacement u(x,t) satisfies the nonlinear KG equation (∂_{t}^{2}-∂_{x}^{2})u=-tan^{-1}u in dimensionless form. Simulations of this equation reveal long-lived, breatherlike states, or "pseudobreathers," which preserve shape and speed to high accuracy when Lorentz-transformed to simulate collisions.