Spectral stability in the modified Camassa-Holm equation

We investigate the spectral stability of small-amplitude, periodic, traveling-wave solutions of the modified Camassa-Holm equation with cubic nonlinearities. More precisely, we analyze the $L^2(\text{mr})$-spectrum of the associated linearized operator in a neighborhood of the origin in the spectral plane. Inspired by a recently novel method based on Kato’s perturbation theory [Berti et al, Full description of Benjamin-Feir instability of Stokes waves in deep water, $\mathit{\text{Invent.}}\mathit{\text{Math.}}$, 230 (2022), 651-711.], we provide a complete description of the spectrum near the origin of the linearized operator–an integro-differential operator with periodic coefficients–and thus prove that such waves are not subject to modulational instability. Moreover, a spectral analysis reveals a remarkable threshold phenomenon: such waves with wave number $k^2\le 3$ exhibit spectral stability, while instability emerges when $k^2>3$.