A Liouville theorem for the critical generalized Korteweg-de Vries equation. (English) Zbl 0963.37058

Martel, Yvan; Merle, Frank

J. Math. Pures Appl. (9) 79, No. 4, 339-425 (2000).

This paper deals with the equation \[ \begin{cases} u_t+(u_{xx}+ u^5)_x= 0,\quad & (t,x)\in\mathbb{R}_+\times\mathbb{R}\\ u(x,0)=u_0(x), \quad x\in\mathbb{R}\end{cases} \tag{1} \] for \(u_0\in H^1(\mathbb{R})\). The authors present a surprising rigidity result on the flow of (1) close to a soliton up to scaling and translation. For a proof of this result the authors introduce new techniques which will give a result of asymptotic completeness.

Reviewer: Messoud Efendiev (Berlin)

MSC:

37K10	Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
35Q53	KdV equations (Korteweg-de Vries equations)
37K40	Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems

Keywords:

Liouville theorem; KdV equation; asymptotic completeness; rigidity