Stability of integrable and nonintegrable structures

In this paper we give a comprehensive account of several recent results on the stability of nontrivial soliton structures for some well-known non periodic dispersive models. We will focus on the simpler case of the generalized Korteweg-de Vries equations, covering the classical stability results by Bona, Souganidis, Strauss until the results by Martel and Merle and our recent collaborations with Miguel Alejo and Luis Vega.