14:30-15:30 Thursday, 10 April 2025 Auditorium, Rectorate
Abstract
The goal of the talk is to discuss the existence of large amplitude traveling waves for the two-dimensional non-resistive Magnetohydrodynamics (MHD) system with an external traveling wave force. More precisely, we assume that the force is a smooth bi-periodic traveling wave propagating in the direction $\omega = (\omega_1, \omega_2)\in \mathbb{R}^2$, with large amplitude of order $O(\lambda^{1^+})$ and large velocity $\lambda\omega$. Then, for most values of $\omega$ and for sufficiently large $\lambda \gg 1$, we construct bi-periodic traveling wave solutions of arbitrarily large amplitude. Due to the presence of small divisors, the proof is based on a nonlinear Nash-Moser scheme adapted to construct nonlinear waves of large amplitude. The main difficulty is that the linearized equation at any approximate solution is an unbounded perturbation of large size of a diagonal operator, and hence the problem is not perturbative. The invertibility of the linearized operator is achieved using tools from micro-local analysis and normal forms, together with a detailed analysis of high- and low-frequency regimes with respect to the large parameter $\lambda$. This is a joint work with Riccardo Montalto (Università di Milano Statale) and Shulamit Terracina (SISSA).