Kink propagation in a two-dimensional curved Josephson junction

We consider the propagation of sine-Gordon kinks in a planar curved strip as a model of nonlinear wave propagation in curved wave guides. The homogeneous Neumann transverse boundary conditions, in the curvilinear coordinates, allow to assume a homogeneous kink solution. Using a simple collective variable approach based on the kink coordinate, we show that curved regions act as potential barriers for the wave and determine the threshold velocity for the kink to cross. The analysis is confirmed by numerical solution of the 2D sine-Gordon equation. PACS numbers: 74.50.+r, 05.45.Yv, 85.25.Cp Recent advances in micro-structuring and nano-structuring technology have made it possible to fabricate various low-dimensional systems with complicated geometry. Examples are photonic crystals with embedded defect structures such as microcavities, wave guides and wave guide bends [1]; narrow constructions (quantum dots and channels) formed at semiconductor heterostructures [2], magnetic nanodisks, dots and rings [3, 4], etc. It is well known that the wave equation subject to Dirichlet boundary conditions has bound states in straight channels of variable width [5] and in curved channels of constant cross-section [6]. Spectral and transport characteristics of quantum electron channels [7] and wave guides in photonic crystal [8] are essentially modified by the existence of segments with finite curvature. Until recently there have been a few theoretical and numerical studies of the effect of curvature on properties of nonlinear excitations. The dynamics of a ring shaped Josephson fluxons and their collisions was studied in [9, 10, 11]. Nonlinear whispering gallery modes for a nonlinear Maxwell equation in microdisks were investigated in [12], the excitation of whispering-gallery-type electromagnetic modes by a moving fluxon in an annular Josephson junction was found in [13]. Nonlinear localized modes in two-dimensional photonic crystal wave