By Jerome Goldstein, Neil Gretsky, John Uhl

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**Example text**

Hence, in the finite dimensional case, we prove a Melnikov theorern for (1) under the non-resonance assumption (38), without the need of the extra assulnption (55). This result is new. From the previous remark, it appears that besides multiple normal frequencies one nlay investigate also the case of norlnal frequencies resonant with the tangential ones, thus when (104), (38) are partially violated. Presently there has been no systematic study of this. (iii) Consider a NLW Btt - ~y + V y + c F' (y) == (105) 0 and denote (106) (assuluing this makes sense).

V] Yau, H. : Logarithmic Sobolev inequality for lattice gases with mixing conditions, preprint.

For instance, perturbations of the KdV-equation Ut + U xxx + UU x == 0 (46) of the form Ut + U xxx + UU x + c j(u)x == 0 (47) (f (u) polynomial or real analytic). The process of extracting the normal form here is rather involved since it is based on the Riemann surface correspondence (cf. [K2]). Coming back to (1), we deal with the problem of persistency of finite dirnensional tori in an infinite dimensional phase space. This is a generalization of the more classical KAM (Kolmogorov-Arnold-Moser) setting of persistency of n tori in 2n-dimensional phase space.