The Gardner equation and the L^2-stability of the N-soliton solution of the Korteweg-de Vries equation

Miguel A. Alejo, Claudio Muñoz, Luis Vega

Published 2010-12-23, updated 2011-01-22Version 3

Multi-soliton solutions of the Korteweg-de Vries equation (KdV) are shown to be globally L2-stable, and asymptotically stable in the sense of Martel-Merle. The proof is surprisingly simple and combines the Gardner transform, which links the Gardner and KdV equations, together with the Martel-Merle-Tsai and Martel-Merle recent results on stability and asymptotic stability in the energy space, applied this time to the Gardner equation. As a by-product, the results of Maddocks-Sachs and Merle-Vega are improved in several directions.