On minimal nonscattering solution for focusing mass-subcritical nonlinear Schrödinger equation

Abstract

We consider time global behavior of solutions to the focusing mass-subcritical NLS equation in weighted L 2 space. We prove that there exists a threshold solution such that (i) it does not scatter; (ii) with respect to a certain scale-invariant quantity, this solution attains minimum value in all non-scattering solutions. In the mass-critical case, it is known that ground states are this kind of threshold solution. However, in our case, it turns out that the above threshold solution is not a standing-wave solution.

Existence of a minimal non-scattering solution to the mass-subcritical generalized Korteweg–de Vries equation

“…Then, it is obvious that (G 1 n ) −1 r 1 n ψ 1 − ψ 1 = 0 weakly inL α as n → ∞. The boundedness (45) for j = 1 is also obvious by (44). By Lemma 5.4,…”

“…Hence, G j n is orthogonal to G k n for 1 k j − 1. Then, by (42) and by Lemma 5.3, we have ψ j ∈ V(u), from which boundedness (44) and (45) follow.…”

“…The proof follows from the argument similar to the proof of Proposition 3.2 or as in [44]. We omit the detail.…”

“…For the focusing mass-critical KdV equation, MartelMerle-Raphaël [41,42,43] and Martel-Merle-Nakanishi-Raphaël [40] classified the dynamics of solution into three cases (blow-up, soliton, away from soliton) in the small neighborhood of Q. As for the mass-subcritical nonlinear Schrödinger equation, the first author treated a minimization problem similar to (1) in a framework of weighted spaces and showed existence of a threshold solution which is smaller than ground state solutions (see [44,45]). …”

Scattering for Stochastic Nonlinear Schrödinger Equations

“…Moreover, global well-posedness and scattering in L 2 are proved in [25] for the masscritical exponent α = 1 + 4/d, d ≥ 3. In the focusing case, there exists a threshold for global well-posedness, scattering and blow-up; we refer to [26,33,34,36,37] and references therein.In the framework of stochastic mechanics, developed by E. Nelson [38], there are also several works devoted to potential scattering, in terms of diffusions instead of wave functions. See, e.g., [12,13,43].However, to the best of our knowledge, there are few results on the scattering problem for stochastic nonlinear Schrödinger equations (1.1).…”