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• W3166731616 abstract "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." @default.
• W3166731616 created "2021-06-22" @default.
• W3166731616 creator A5042267493 @default.
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• W3166731616 date "2021-06-14" @default.
• W3166731616 modified "2023-10-17" @default.
• W3166731616 title "Long Time Dynamics for Generalized Korteweg–de Vries and Benjamin–Ono Equations" @default.
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• W3166731616 doi "https://doi.org/10.1007/s00205-021-01666-z" @default.
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