ASYMPTOTIC STABILITY OF KDV SOLITONS ON THE HALF-LINE: A STUDY IN THE ENERGY SPACE.

In this paper we study the asymptotic stability problem for KdV solitons on the half-line, with zero boundary condition and absence of the drift term, represented as ux. Unlike standard KdV, these are not exact solutions to the equation. In a previous result, we showed that these solitons are orbitally stable, provided they are placed sufficiently far from the origin. In this paper, we prove their asymptotic stability in the energy space, and provide decay properties for all remaining regions, except the "small soliton region". For the proof we follow the ideas by Martel and Merle for the big soliton part, and for the linearly dominated region we follow recent results on generalized KdV decay [C. Muñoz and G. Ponce, Comm. Math. Phys., 367 (2019), pp. 581-598, A. [ABSTRACT FROM AUTHOR]