Modulational instability of small amplitude periodic traveling waves in the $b$-family of Novikov equation

We study the modulational instability of smooth, small-amplitude periodic traveling wave solutions to the $b$-family of Novikov equation with cubic nonlinearity with an arbitrary coefficient $b>0$. Our approach is based on applying spectral perturbation theory to the corresponding linearization process. We derive a modulation instability index dependent on the nonlinear parameter $b$ and the fundamental wave number, and prove that when this index is negative, sufficiently small periodic traveling waves in the Novikov equation $b$-family exhibit spectral instability to long-wavelength perturbations. This confirms the well-known Benjamin-Feir instability in the $b$-family of Novikov equation.