Narrow Results

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 with respect 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 result of large amplitude quasi-periodic solutions for a nonlinear PDE in higher space dimension.