F. Natali, D.E. Pelinovsky, and S. Wang
Stability of periodic waves in the model with intensity-dependent dispersion
We study standing periodic waves modeled by the nonlinear Schrodinger
equation with the intensity-dependent dispersion coecient. Spatial periodic profiles are
smooth if the frequency of the standing waves is below the limiting frequency, for which
the profiles become peaked (piecewise continuously di erentiable with a finite jump of
the first derivative). We prove that there exist two families of the periodic waves with
smooth profiles separated by a homoclinic orbit and the period function (the energy-toperiod
mapping) is monotonically increasing for the family inside the homoclinic orbit
and decreasing for the family outside the homoclinic orbit. This property allows us to
derive a sharp criterion for the energetic stability of such standing periodic waves under
time evolution if the perturbations are periodic with the same period for both families
and, additionally, for the family outside the homoclinic orbit, spatially odd with respect
to the half-period. By numerically approximating the sharp stability criterion, we show
that both families are energetically stable for small frequencies but become unstable
when the frequency approaches the limiting frequency of the peaked waves.
Keywords:
nonlinear Schrodinger equation, intensity-dependent dispersion, standing periodic waves, spectal stability,
period function, orbital stability.