We study the stability and dynamics of solitons in the Korteweg-de Vries
(KdV) equation in the presence of noise and deterministic forcing. The noise is
space-dependent and statistically translation-invariant. We show that, for
small forcing, solitons remain close to the family of traveling waves in a
weighted Sobolev norm, with high probability. We study the effective dynamics
of the soliton amplitude and position via their variational phase, for which we
derive explicit modulation equations. The stability result holds on a time
scale where the deterministic forcing induces significant amplitude modulation.

Cet article explore les excursions dans le temps et leurs implications.

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