High frequency perturbation of cnoidal waves in KdV
  The Korteweg-de Vries (KdV) equation with periodic boundary conditions is considered. The interaction of a periodic solitary wave (cnoidal wave) with high frequency radiation of finite energy ($L^2$-norm) is studied. It is proved that the interaction of low frequency component (cnoidal wave) and high frequency radiation is weak for finite time in the following sense: the radiation approximately satisfies Airy equation. 
M.B. Erdo\u{g}an, N. Tzirakis, V. Zharnitsky
math.AP math-ph math.MP