Long time gyrokinetic equations

The aim of this text is to elucidate the oscillating patterns (see C. Cheverry [Res. Rep. Math. (2018)]) which are generated in a toroidal plasma by a strong external magnetic field and a nonzero electric field. It is also to justify and then study new modulation equations which are valid for longer times than before. Oscillating coherent structures are induced by the collective motions of charged particles which satisfy a system of ODEs implying a large parameter, the gyrofrequency ε − 1 ≫ 1 \varepsilon ^{-1} \gg 1 . By exploiting the properties of underlying integrable systems, we can complement the KAM picture (see G. Benettin and P. Sempio [Nonlinearity 7 (1994), pp. 281–303]; M. Braun [SIAM Rev. 23 (1981), pp. 61–93]) and go beyond the classical results about gyrokinetics (see M. Bostan [Multiscale Model. Simul. 8 (2010), pp. 1923–1957]; A. J. Brizard and T. S. Hahm [Rev. Modern Phys. 79 (2007), pp. 421–468]). The purely magnetic situation was addressed by C. Cheverry [Comm. Math. Phys. 338 (2015), pp. 641–703; J. Differential Equations 262 (2017), pp. 2987–3033]. We are concerned here with the numerous additional difficulties due to the influence of a nonzero electric field.