Some results on the scattering of weakly interacting solitons for nonlinear Schrödinger equations. (English) Zbl 0931.35164

Demuth, Michael (ed.) et al., Spectral theory, microlocal analysis, singular manifolds. Berlin: Akademie Verlag. Math. Top. 14, 78-137 (1997).
The asymptotic behaviour of solutions of the Cauchy problem for the equation \[ i\psi_t=-\psi_{xx}+F\bigl( |\psi|^2 \bigr)\psi,x,t\in \mathbb{R}\tag{1} \] with initial data close to a superposition of two solitons is studied. Sufficient conditions on the function \(F\) and on the linearizations of the equation at the solitons, in order that \(\psi\) be asymptotically for \(t\to+ \infty\) equal with the sum of two solitons with slightly deformed parameters and of a small term, which satisfies the free Schrödinger equation, are given. It is also proved that if the solution of the equation (1) is asymptotically for \(t \to-\infty\) equal with the superposition of two solitons, then this conclusion is still valid. Similar results for the case of a single soliton were previously proved by the author and V. S. Buslaev [see e.g. V. G. Buslaev, G. S. Perelman, Asterisque, 210, 49-63 (1992; Zbl 0795.35111)].
For the entire collection see [Zbl 0882.00015].


35Q55 NLS equations (nonlinear Schrödinger equations)
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
35P25 Scattering theory for PDEs


Zbl 0795.35111