Large amplitude traveling waves for the non-resistive MHD system

Abstract

We prove the existence of large amplitude bi-periodic traveling waves (stationary in a moving frame) of the two-dimensional non-resistive Magnetohydrodynamics (MHD) system with a traveling wave external force with large velocity speed $\lambda (\omega_1, \omega_2)$ and of amplitude of order $O(\lambda^{1^+})$ where $\lambda \gg 1$ is a large parameter. For most values of $\omega = (\omega_1, \omega_2)$ and for $\lambda \gg 1$ large enough, we construct bi-periodic traveling wave solutions of arbitrarily large amplitude as $\lambda \to + \infty$. More precisely, we show that the velocity field is of order $O(\lambda^{0^+})$, whereas the magnetic field is close to a constant vector as $\lambda \to + \infty$. 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 then performed by using tools from micro-local analysis and normal forms together with a sharp analysis of high and low frequency regimes w.r. to the large parameter $\lambda \gg 1$. To the best of our knowledge, this is the first result in which global in time, large amplitude solutions are constructed for the 2D non-resistive MHD system with periodic boundary conditions and also the first existence results of large amplitude quasi-periodic solutions for a nonlinear PDE in higher space dimension.