\(L^2\) stability of solitons for KdV equation. (English) Zbl 1022.35061

Merle, F.; Vega, L.

Int. Math. Res. Not. 2003, No. 13, 735-753 (2003).

The paper revisits the classical problem of the stability of solitons in the Korteweg - de Vries (KdV) equation with the quadratic nonlinearity. It was earlier rigorously proved that solitons, as solutions to the KdV equation, are stable and asymptotically stable in the \(H^1\) functional space. The aim of the present paper is to give a strict proof of the stability and asymptotic stability of the solitons in the \(L^2\) space. The proof is based on using the Miura transformation, which relates the KdV equation to the modified KdV equation (with cubic nonlinearity, rather than quadratic one). In the modified KdV equation, the KdV soliton corresponds to a kink (topological soliton), whose stability and asymptotic stability in the \(L^2\) space have been proved before.

Reviewer: Boris A.Malomed (Tel Aviv)

MSC:

35Q53	KdV equations (Korteweg-de Vries equations)
37K45	Stability problems for infinite-dimensional Hamiltonian and Lagrangian systems

Keywords:

modified KdV equation; stability; Miura transformation; asymptotic stability; kink; topological soliton; KdV equation