On nonlinear scattering of states which are close to a soliton. (English) Zbl 0795.35111

Robert, D. (ed.), Méthodes semi-classiques, Volume 2. Colloque international (Nantes, juin 1991). Paris: Société Mathématique de France, Astérisque. 210, 49-63 (1992).
The authors report on their recent results on the large time behavior of solutions of \(i\psi_ t+ \psi_{xx}= F(|\psi|^ 2)\psi\) with initial value close to a family of exact solutions which generalize the one-soliton for the cubic nonlinear Schrödinger equation. They find in particular that if \(F\) is a polynomial, vanishes to fourth order at 0, and is bounded below, the solution tends, in \(L^ 2\), to the sum of a member of the family and a solution of the linear Schrödinger equation, modulo an additional condition on the linearization of the equation. In this sense, one can thus define wave operators at \(+\infty\).
The method of proof is outlined and compared with the work of Soffer and M. Weinstein on nonlinear perturbations of linear Schrödinger equations, where the role of the one-soliton is played by an eigenfunction of the unperturbed problem.
For the entire collection see [Zbl 0778.00035].


35Q55 NLS equations (nonlinear Schrödinger equations)
35P25 Scattering theory for PDEs
35B40 Asymptotic behavior of solutions to PDEs