S. Cui and D.E. Pelinovsky
Instability bands for periodic traveling waves in the modified Korteweg-de Vries equation
Abstract:
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. The new discovery of this work is 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-infinity (crossing at the real axis).
Keywords:
modified Korteweg-de Vries equation, periodic traveling waves, spectral stability, squared
eigenfunctions, Lax spectrum, figure-8 instability, figure-infinity instability