Space- and time-periodic perturbations of the Sine-Gordon equation

Abstract

We study the Sine-Gordon equation ϕtt - ϕzz + sinϕ = 0 subject to two classes of small perturbations: (1) spatially periodic perturbations on an infinite domain and (2) weak dissipation and temporally periodic perturbations on the boundary of a finite spatial domain. In the former case we prove the existence of a countable set of spatially periodic stationary solutions in addition to an uncountable set of non-periodic stationary solutions. In the latter case we prove that, if the excitation is sufficiently large compared with dissipation, a countable set of time-periodic motions of all periods as well as non periodic motions exist. All these stationary and periodic solutions are unstable (of saddle type) and are expected to coexist with stable solutions. However, in the global bifurcations leading to creation of the latter (time-dependent) solutions, infinite sets of asymptotically stable periodic orbits with arbitrarily long periods are expected to appear.