Stability of Travelling Wave Solutions to the Sine-Gordon Equation

2010/10/13 by Jones, C. K. R. T., Marangell, R.
#35P99 #47A75 #47J10 #Analysis of PDEs (math.AP) #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.1010.2759

Abstract

We give a geometric proof of spectral stability of travelling kink wave solutions to the sine-Gordon equation. For a travelling kink wave solution of speed c ≠ ± 1, the wave is spectrally stable. The proof uses the Maslov index as a means for determining the lack of real eigenvalues. Ricatti equations and further geometric considerations are also used in establishing stability.

Related

• Asymptotic stability of the sine-Gordon kink
• Asymptotic stability of the sine-Gordon kinks under perturbations in weighted Sobolev norms
• On the spectral stability of kinks in some PT-symmetric variants of the classical Klein-Gordon Field Theories
• On nonlinear wave equations with parabolic potentials
• Nonlinear stability of 2-solitons of the Sine-Gordon equation in the energy space