Asymptotic stability of solitons for subcritical generalized KdV equations. (English) Zbl 0981.35073

Martel, Yvan; Merle, Frank

Arch. Ration. Mech. Anal. 157, No. 3, 219-254 (2001).

This paper is devoted to the generalized Korteweg-de Vries equation in the subcritical case, that is \[ \begin{cases} u_t+ (u_{xx}+ u^p)_x= 0,\quad &(t,x)\in \mathbb{R}\times \mathbb{R},\\ u(0, x)= u_0(x),\quad & x\in\mathbb{R}\end{cases}\tag{1} \] for \(p= 2,3,4\) and \(u_0\in H^1(\mathbb{R})\). The author proves asymptotic completeness of the family of solutions in the energy space for (1).

Reviewer: Messoud Efendiev (Berlin)

MSC:

35Q53	KdV equations (Korteweg-de Vries equations)
37K45	Stability problems for infinite-dimensional Hamiltonian and Lagrangian systems
35B35	Stability in context of PDEs
35B40	Asymptotic behavior of solutions to PDEs

Keywords:

generalized Korteweg-de Vries equation; subcritical case; asymptotic completeness; energy space