Instability bands for periodic traveling waves in the modified Korteweg-de Vries equation

Two families of periodic traveling waves exist in the focusing mKdV (modified Korteweg-de Vries) equation. Spectral stability of these waveforms with respect to co-periodic perturbations of the same period has been previously explored by using spectral analysis and variational formulation. By using tools of integrability such as a relation between squared eigenfunctions of the Lax pair and eigenfunctions of the linearized stability problem, we revisit the spectral stability of these waveforms with respect to perturbations of arbitrary periods. In agreement with previous works, we find that one family is spectrally stable for all parameter configurations, whereas the other family is spectrally unstable for all parameter configurations. We show that the onset of the co-periodic instability for the latter family changes the instability bands from figure-$8$ (crossing at the imaginary axis) into figure-$\infty$ (crossing at the real axis).