On the stability of KdV multi‐solitons

We consider the stability of multi‐ or n ‐soliton solutions to the Korteweg‐de Vries equation (KdV) posed on the real line. It is shown that in the standard variational characterization of KdV multi‐solitons as critical points, the n ‐solitons actually realize non‐isolated constrained minimizers. (The case n = 1 was already known to Benjamin; see [6].) From this fact a precise dynamic stability result for multi‐solitons follows, namely, that initial data close to a given n ‐soliton evolves in time so as to remain close (in the H n (𝔑) Sobolev norm) to the n ‐dimensional manifold of all n ‐solitons with appropriate wave speeds, i.e., to the set of constrained minimizers. Our techniques are also applicable to other Hamiltonian systems with several conserved quantities. In particular the inverse scattering formalism of KdV is not explicitly exploited.