The Gardner equation and the 𝐿²-stability of the 𝑁-soliton solution of the Korteweg-de Vries equation

Abstract

Multi-soliton solutions of the Korteweg-de Vries equation (KdV) are shown to be globally L 2 L^2 -stable, and asymptotically stable in the sense of Martel and 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.