F. Natali, U. Le and D.E. Pelinovsky
New variational characterization of periodic waves in the fractional Korteweg-de Vries equation
Nonlinearity 33 (2020) 1956-1986
Abstract:
Periodic waves in the fractional Korteweg - de Vries equation have been previously
characterized as constrained minimizers of energy subject to fixed momentum and mass. Here
we characterize these periodic waves as constrained minimizers of the quadratic form of energy
subject to fixed cubic part of energy and the zero mean. This new variational characterization
allows us to unfold the existence region of travelling periodic waves and to give a sharp criterion
for spectral stability of periodic waves with respect to perturbations of the same period. The
sharp stability criterion is given by the monotonicity of the map from the wave speed to the
wave momentum similarly to the stability criterion for solitary waves.
Keywords:
Fractional Korteweg-de Vries equation, traveling periodic waves, varatiational characterization,
stability of periodic waves.