arXiv subjects

Charbella Abou Khalil

Publications and source records attributed to Charbella Abou Khalil.

2 recordsLinked to original sources

arXiv2025

Almost conservation of the harmonic actions for fully discretized nonlinear Klein--Gordon equations at low regularity

Close to the origin, the nonlinear Klein--Gordon equations on the circle are nearly integrable Hamiltonian systems which have infinitely many almost conserved quantities called harmonic actions or super-actions. We prove that, at low regularity and with a CFL number of size 1, this property is preserved if we discretize the nonlinear Klein--Gordon equations with the symplectic mollified impulse methods. This extends previous results of D. Cohen, E. Hairer and C. Lubich to non-smooth solutions.

math.AP↗

arXiv2024

Birkhoff normal form in low regularity for the nonlinear quantum harmonic oscillator

Given small initial solutions of the nonlinear quantum harmonic oscillator on $\mathbb{R}$, we are interested in their long time behavior in the energy space which is an adapted Sobolev space. We perturbate the linear part by $V$ taken as multiplicative potentials, in a way that the linear frequencies satisfy a non-resonance condition. More precisely, we prove that for almost all potentials $V$, the low modes of the solution are almost preserved for very long times.

math.AP↗