Stability of the multi-solitons of the modified Korteweg–de Vries equation ^*

We establish the nonlinear stability of $N$-soliton solutions modified Korteweg-de Vries (mKdV) equation. The are global mKdV behaving at (positive and negative) time infinity as sums $1$-solitons with speeds $0<c_1<\cdots< c_N$.The proof relies on variational characterization $N$-solitons. show that $N$-solitons realize local minimum $(N+1)$-th conserved quantity subject to fixed constraints $N$ first quantities.To this aim, we construct a functional for which critical points, prove spectral properties linearization around preserved extended timeline, analyze spectrum linearized operators $1$-solitons. main new ingredients in our analysis operator identity based generalized Sylvester law inertia recursion