THE GARDNER EQUATION AND THE STABILITY OF MULTI-KINK SOLUTIONS OF THE MKDV EQUATION

Multi-kink solutions of the defocusing, modified Korteweg-de Vries equation (mKdV) found by Grosse [22, 23] are shown to be globally H-1-stable, and asymptotically stable. Stability in the one-kink case was previously established by Zhidkov [51] and Merle-Vega [41]. The proof uses transformations linking the mKdV equation with focusing, Gardner-like equations, where stability and asymptotic stability in the energy space are known. We generalize our results by considering the existence, uniqueness and the dynamics of generalized multi-kinks of defocusing, non-integrable gKdV equations, showing the inelastic character of the kink-kink collision in some regimes.