| Abstract: |
| The goal of this talk is to discuss the existence of large amplitude traveling waves of the two-dimensional nonresistive Magnetohydrodynamics (MHD) system with a traveling wave external 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 with large velocity speed $\lambda\omega$.
Then, for most values of $\omega$ and for $\lambda\gg1$ large enough,
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 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 $\lambda$.
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This is a joint work with G. Ciampa and R. Montalto. |
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