Stability of the multi-solitons of the modified Korteweg-de Vries equation

We establish the nonlinear stability of $N$-soliton solutions of the modified Korteweg-de Vries (mKdV) equation. The $N$-soliton solutions are global solutions of mKdV behaving at (positive and negative) time infinity as sums of $1$-solitons with speeds 0 < c1 < · · · < cN . The proof relies on the variational characterization of $N$-soliton. We show that the $N$-solitons realize the local minimum of the $(N+1)$-th mKdV conserved quantity subject to fixed constraints on the $N$ first conserved quantities. To this aim, we construct a functional for which $N$-solitons are critical points, we prove that the spectral properties of the linearization of this functional around a $N$-soliton are preserved on the extended timeline, and we analyze the spectrum at infinity of linearized operators around $1$-solitons.The main new ingredients in our analysis are a new operator identity based on a generalized Sylvester law of inertia and recursion operators for the mKdV equation.