## Abstract

Consider a weakly nonlinear CGL equation on the torus T^{d}:\begin{aligned} (Formula Presented.) N. Define I(u) = (I_{k}, k∈ Z^{d}) , where I_{k}= v_{k}v¯ _{k}/ 2 and v_{k}, k∈ Z^{d}, are the Fourier coefficients of the function u we give. Assume that the equation (∗) is well posed on time intervals of order ϵ^{- 1} and its solutions have there a-priori bounds, independent of the small parameter. Let u(t, x) solve the equation Faou, E., Germain, P., Hani, Z.: The weakly nonlinear large box limit of the 2d cubic nonlinear Schrödinger equation. preprint, 2013.∗). If ϵ is small enough, then for t≲ ϵ^{- 1}, the quantity I(u(t, x)) can be well described by solutions of an effective equation: (Formula Presented.),where the term F(u) can be constructed through a kind of resonant averaging of the nonlinearity b|u| ^{2}^{p}+ ic| u| ^{2}^{q}u.

Original language | English |
---|---|

Pages (from-to) | 375-387 |

Number of pages | 13 |

Journal | Journal of Dynamics and Differential Equations |

Volume | 28 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1 Jun 2016 |

Externally published | Yes |