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

我们提供了一维环面上的广义 Korteweg-De Vries (gKdV) 和 Benjamin-Ono (gBO) 方程解的长时间动态的准确描述，没有外部参数，并且几乎所有（在概率和密度）小而平滑的初始数据。特别是，我们证明了 Sobolev 范数的长期稳定性结果：给定一个大常数r和一个足够小的参数\(\varepsilon \)，对于大小为\(\varepsilon \) 的通用初始数据u (0 )，我们控制解u ( t )的 Sobolev 范数的阶次\(\varepsilon ^{-r}\). 这些结果是通过将系统置于有理规范形式中获得的：我们将这些方程的向量场共轭到一些高阶余数项，以高 Sobolev 正则性将这些方程的矢量场在围绕原点的大开集上可积。我们强调，我们的范式技术首次允许处理包含偶数阶项的无界非线性。