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

TL;DR

This paper investigates how small diffusion terms affect the solutions of the perturbed sine-Gordon equation with Neumann boundary conditions, relevant to Josephson junctions, showing diffusion effects diminish as the diffusion coefficient approaches zero.

Contribution

It provides a detailed analysis of the diffusion effects in the perturbed sine-Gordon equation, including the Green function construction and solution boundedness, which was not previously established.

Findings

Diffusion effects are bounded and vanish as the diffusion coefficient approaches zero.

A Green function for the linear third order operator is explicitly constructed.

Existence of solutions with bounded derivatives is proven.

Abstract

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.