On diffusion effects of the perturbed sine-Gordon equation with Neumann boundary conditions

The Neumann boundary problem for the perturbed sine-Gordon equation describing the electrodynamics of Josephson junctions has been considered. The behavior of a viscous term, described by a higher-order derivative with small diffusion coefficient, is investigated. The Green function related to the linear third order operator is determined by means of Fourier series, and properties of rapid convergence are established. Furthermore, some classes of solutions of the hyperbolic equation have been determined, proving that there exists at least one solution whose derivatives are bounded. Results prove that diffusion effects are bounded and tend to zero when the small diffusion coefficient tends to zero.