Spectral Stability of Travelling Waves in a δ-Regularized Dissipative Sine-Gordon Equation

We analyze the spectral stability of travelling waves in a δ-regularized dissipative sine-Gordon equation modelling refined long Josephson junction dynamics. Linearization about a wave yields a singularly perturbed fourth-order spectral problem with intrinsic slow–fast spatial structure. Using an Evans-function formulation on a domain of consistent spatial splitting, we establish a local factorization separating slow and fast modes and prove that the δ-induced fast subsystem remains uniformly hyperbolic and does not generate an additional point spectrum near λ=0. Hence, the local point spectrum coincides with that of the classical dissipative sine-Gordon equation. Numerical computations of the essential spectrum and Evans winding numbers confirm the analysis and show that the higher-order terms enhance high-frequency damping without altering low-frequency spectral stability.