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Stable manifolds for all monic supercritical focusing nonlinear Schrödinger equations in one dimension. (English) Zbl 1281.35077

Summary: Standing wave solutions of the one-dimensional nonlinear Schrödinger equation \[ \partial_t \psi + \partial_{x}^2 \psi =-|\psi|^{2\sigma} \psi \] with \(\sigma>2\) are well known to be unstable. We show that asymptotic stability can be achieved provided the perturbations of these standing waves are small and chosen to belong to a codimension one Lipschitz surface. Thus, we construct codimension one asymptotically stable manifolds for all supercritical NLS in one dimension. The considerably more difficult \(L^2\)-critical case, for which one wishes to understand the conditional stability of the pseudo-conformal blow-up solutions, is studied in the authors’ companion paper [J. Eur. Math. Soc. (JEMS) 11, No. 1, 1–125 (2009; Zbl 1163.35035)].

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35Q51 Soliton equations
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
37K45 Stability problems for infinite-dimensional Hamiltonian and Lagrangian systems

Citations:

Zbl 1163.35035
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