On kinks and other travelling-wave solutions of a modified sine-Gordon equation

Abstract

We give an exhaustive, non-perturbative classification of exact travelling-wave solutions of a perturbed sine-Gordon equation (on the real line or on the circle) which is used to describe the Josephson effect in the theory of superconductors and other remarkable physical phenomena. The perturbation of the equation consists of a constant forcing term and a linear dissipative term. On the real line candidate orbitally stable solutions with bounded energy density are either the constant one, or of kink (i.e. soliton) type, or of array-of-kinks type, or of "half-array-of-kinks" type. While the first three have unperturbed analogs, the last type is essentially new. We also propose a convergent method of successive approximations of the (anti)kink solution based on a careful application of the fixed point theorem.

arXiv commentLatex file, 25 pages, 4 figures. Final version to appear in "Meccanica"

References in corpus (5)

• On the stability analysis of periodic sine-Gordon traveling waves
• Existence and uniqueness of solutions of a class of 3rd order dissipative problems with various boundary conditions describing the Josephson effect
• Towards soliton solutions of a perturbed sine-Gordon equation
• Global Stability properties for a class of dissipative phenomena via one or several Liapunov functionals
• Existence, uniqueness and stability for a class of third order dissipative problems depending on time

Cited by in corpus (6)

• Josephson Flux Flow Oscillator: the Microscopic Tunneling Approach
• A scattering theory of ultrarelativistic solitons
• Head butting sheep: Kink Collisions in the Presence of False Vacua
• On the transition from parabolicity to hyperbolicity for a nonlinear equation under Neumann boundary conditions
• Mathematical Contributions to the Dynamics of the Josephson Junctions: State of the Art and Open Problems
• On localized and coherent states on some new fuzzy spheres