Long Time Dynamics for Generalized Korteweg–de Vries and Benjamin–Ono Equations

We provide an accurate description of the long time dynamics solutions generalized Korteweg–De Vries (gKdV) and Benjamin–Ono (gBO) equations on one dimension torus, without external parameters, that are issued from almost any (in probability in density) small smooth initial data. In particular, we prove a long-time stability result Sobolev norm: given large constant r sufficiently parameter $$\varepsilon $$ , for generic datum u(0) size control norm solution u(t) times order ^{-r}$$ . These results obtained by putting system rational normal form: conjugate, up to some high remainder terms, vector fields these integrable ones open sets surrounding origin regularity. stress out our form technics allow deal, first time, with unbounded nonlinearities containing terms even order.