Orbitally stable standing waves for the asymptotically linear one-dimensional NLS

Abstract

In this article we study the one-dimensional, asymptotically linear, non-linear Schrödinger equation (NLS). We show the existence of a global smooth curve of standing waves for this problem, and we prove that these standing waves are orbitally stable. As far as we know, this is the first rigorous stability result for the asymptotically linear NLS. We also discuss an application of our results to self-focusing waveguides with a saturable refrac-1

Orbital Stability: Analysis Meets Geometry

“…This has been proved in [Gen10a] for the (PT) case and in [Gen13] for the (AL) case. The proofs rely on the theory of orbital stability in [GSS87] and we will now outline the main arguments.…”

“…In fact, the case where asymptotic bifurcation occurs at ξ = 0, corresponding in dimension d = 1 to 5 − 2b < σ < ∞, could also be extended to the (AL) case, where instability could be inferred, in the limit ξ → 0. We refrain from going in this direction here since we were only able so far to extend the 18 More general assumptions on the coefficient f in (AL) can be given, under which the bifurcation and stability results presented here still hold, see [Gen13].…”

“…In the setting of Section 8, they can be seen as an application of Theorem 8.6. The approach used in [GS08,Gen09,Gen10a,Gen13] was to apply the celebrated Theorem 2 of Grillakis, Shatah, Strauss [GSS87]. This result essentially relies on the set of spectral conditions (S1)-(S3), formulated below in the context of (10.1), together with a convexity condition, which here takes the form (10.14).…”

“…with f (x, w 2 ) = V (x) |w| σ−1 1+|w| σ−1 ). We will give a short account of the main arguments used in [GS08,Gen09,Gen10a,Gen13] to establish the stability of standing waves along a global solution curve. We will also briefly sketch the bifurcation analysis yielding a smooth branch of non-trivial solutions of (10.2) emerging from the trivial solution w = 0.…”

“…Such standing waves are sometimes referred to as "solitons" due to the spatial localization of the profile w(x), and to their stability. Second, constructing curves of standing wave solutions of (1.5) is now a hard problem, and we will outline the bifurcation theory developed in [GS08,Gen09,Gen10a,Gen13] to solve it. This powerful approach allows one to deal with power-type nonlinearities f (x, |u| 2 ) = V (x)|u| σ −1 (under an approriate decay assumption on the coefficient V : R d → R) but also with more general nonlinearities, for instance the asymptotically linear f (x, |u| 2 ) = V (x) |u| σ−1 1+|u| σ−1 .…”