Stability of the multi-solitons of the modified Korteweg–de Vries equation <sup>*</sup>
Stefan Le Coz, Zhong Wang
Abstract
Abstract 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 one-solitons with speeds 0 < c 1 <…< c N . The proof relies on the variational characterization of N -solitons. 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 an N -soliton are preserved on the extended timeline, and we analyze the spectrum at infinity of linearized operators around one-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.