Spectral analysis of small amplitude periodic $b$-Novikov equation under transverse perturbations

This paper is devoted to the transverse stability problem for small-amplitude periodic traveling waves of the $b$-Novikov equation, which arises as a two-dimensional extension of the $b$-family hierarchy. The perturbations under consideration fall into two groups. One group matches the fundamental period of the underlying wave along the longitudinal direction, while the other consists of profiles that are bounded or localized in the transverse direction. We perform spectral analysis of the associated linearized operator and derive precise stability and instability thresholds. The resulting conditions exhibit intricate dependence on two key parameters including the cubic coupling strength $b$ and the wavenumber $k$.