We construct modified energies for the generalized KdV equation. As a consequence, we obtain quasi-invariance of the high-order Gaussian measures, along with regularity on the corresponding Radon–Nikodym density, as well as new bounds on the growth of the Sobolev norms of the solutions.
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Fabrice Planchon, Nikolay Tzvetkov, Nicola Visciglia, Modified energies for the periodic generalized KdV equation and applications. Ann. Inst. H. Poincaré Anal. Non Linéaire 40 (2023), no. 4, pp. 863–917DOI 10.4171/AIHPC/62