Linear Problems Related to Asymptotic Stability of Solitons of the Generalized KdV Equations

This paper is concerned with the classification of global and uniformly localized solutions of linear problems related to the dynamics of the generalized Korteweg–de Vries equations in a neighborhood of solitons. In this context, we give a simpler and more general proof of a classification result introduced by Martel and Merle J. Math. Pures Appl., 79 (2000), pp. 339–425, Arch. Ration. Mech. Anal., 157 (2001), pp. 219–254. We also consider the case of a nonlinear equation similar to the Korteweg–de Vries equation which was introduced by Peregrine J. Fluid Mechanics, 27 (1967), pp. 815–827 and Benjamin, Bona, and Mahony Philos. Trans. Roy. Soc. London Ser., 272 (1972), pp. 47–78.