On small energy stabilization in the NLS with a trapping potential

Cuccagna, Scipio; Maeda, Masaya

doi:10.2140/apde.2015.8.1289

Cited by 37 publications

(143 citation statements)

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“…Unfortunately, for the cases N 0 = 2, 3, G will be 0 (and so Γ = 0) due to the fact that the nonlinearity has no cubic and quintic term. However, one can still assume a generalized version of Fermi Golden Rule assumption such as [11] and obtain the same result in Theorem 1.9 as well as Theorems 1.15, 1.16 with some modification of the proof. In these case, we will have to take into account the higher order terms and in particular, G appearing in (1.12)-(1.14) will have to be modified as…”

Section: )mentioning

confidence: 78%

“…Therefore, one can expect that for the continuous NLS, all small solutions decouple into a nonlinear bound state and dispersive wave (and in particular no small quasi-periodic solution exists). Indeed, for NLS on R 3 this was shown by Soffer-Weinstein [42] and Tsai-Yau [45] for the two eigenvalue cases with N 0 = 2 and Cuccagna-Maeda [11] for the general cases. Therefore, our result in this paper is similar to the continuous NLS (for related results for nonlinear Klein-Gordon and Dirac equations, see [13] and [5,15,38]).…”

Section: Introductionmentioning

confidence: 74%

“…The proof of Theorems 1.9, 1.15 and 1.16 are based on the argument developed in [11]. Following standard arguments, we first decompose the solution in the form u(t) = φ 1 (z 1 (t)) + φ 2 (z 2 (t)) + η, where η satisfying suitable orthogonal conditions.…”

Section: )mentioning

confidence: 99%

“…This is done by Darboux theorem (Proposition 3.1). We next apply the Birkhoff normal form argument (Proposition 3.3), which is another change of coordinate, developed in [2,3,8,9,10,11]. By Birkhoff normal form argument, we can change the coordinate (z 1 , z 2 , η) s.t.…”

Section: )mentioning

confidence: 99%

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Stabilization of small solutions of discrete NLS with potential having two eigenvalues

Maeda

2019

Applicable Analysis

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We study the long time behavior of small (in l 2 ) solutions of discrete nonlinear Schrödinger equations with potential. In particular, we are interested in the case that the corresponding discrete Schrödinger operator has exactly two eigenvalues. We show that under the nondegeneracy condition of Fermi Golden Rule, all small solutions decompose into a nonlinear bound state and dispersive wave. We further show the instability of excited states and generalized equipartition property.We are interested in the long time behavior of general small solutions of DNLS (1.1). By small solutions, we mean solutions of (1.1) with initial data u(0, ·) = u 0 ∈ l 2 with u 0 2 l 2 := n∈Z |u 0 (n)| 2 sufficiently small. Notice that by the potential V , the discrete Schrödinger operator H = −∆ + V may have eigenvalues. In this case one can show that there exist nonlinear bound states associated to the eigenvalues of H. Here, a nonlinear bound state is a solution of DNLS (1.1) with the form e −iωt φ ω (n) (see Proposition 1.4. Further, for other types of nonlinear bound states see [1]).When H has no eigenvalues, it is known that all small (in l 2 ) solutions scatter. By scattering, we mean that there exists η + ∈ l 2 s.t. the solution converges (in l 2 ) to the free solution e it∆ η + as t → ∞. For the case V ≡ 0 this was shown by Stefanov-Kevrekidis [43]. For the case V = 0, it follows from the dispersive estimate of H proved by Pelinovsky-Stefanov [37] (see also [27] and for lower power nonlinearity case, see [31]). However, we do not know an example s.t. V = 0 and −∆ + V has no eigenvalues (see section 4 and appendix of [27]).When H has one eigenvalue, it is known that all small solutions decouple into a nonlinear bound state and dispersive wave. This means that after subtracting suitable nonlinear bound state from the solution, the remainder scatters. Therefore, the solution u(t) can be expressed as

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Section: )mentioning

confidence: 78%

Section: Introductionmentioning

confidence: 74%

Section: )mentioning

confidence: 99%

Section: )mentioning

confidence: 99%

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Stabilization of small solutions of discrete NLS with potential having two eigenvalues

Maeda

2019

Applicable Analysis

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“…These were obtained for the first time in [20] for the NLS equation with pure power nonlinearity, that is β(|u|) = |u| p , for 0 < p < 4/(d − 2) and were successively used for proving the orbital and asymptotic stability by various authors such as in [3] and [5] for the NLS general local nonlinearities. Recently, non-local nonlinearities in (1.1) where took into account.…”

Section: ) and Ignoring The O(ϵ)-terms It Is Possible To Obtain The mentioning

confidence: 99%