On small energy stabilization in the NLKG with a trapping potential

Cuccagna, Scipio; Maeda, Masaya; Phan, Tuoc

doi:10.1016/j.na.2016.08.009

Cited by 11 publications

(10 citation statements)

References 39 publications

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“…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]). For experimental realization, see [30].…”

Section: Introductionsupporting

confidence: 80%

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: Introductionsupporting

confidence: 80%

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

Maeda

2019

Applicable Analysis

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“…The case of the nonlinear Klein Gordon equation (NLKG), in the context of real valued solutions, where all small solutions scatter to 0, has been initiated in [47] in special case and to a large degree solved in a general way in [2]. For complex valued solutions of the NLKG see [22].…”

Section: Introductionmentioning

confidence: 99%

On stabilization of small solutions in the nonlinear Dirac equation with a trapping potential

Cuccagna

Tarulli

2016

Journal of Mathematical Analysis and Applications

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“…Let us mention the results on convergence of small solutions to (one-frequency) solitary waves, particularly in the context of the nonlinear Schrödinger equation: in other words, the attractor of small solutions is formed by small amplitude solitary waves. See in particular [TY02,SW04,CM15,CMP16,CT16].…”

Section: Introductionmentioning

confidence: 99%

Solutions with Compact Time Spectrum to Nonlinear Klein–Gordon and Schrödinger Equations and the Titchmarsh Theorem for Partial Convolution

Comech

2019

Arnold Math J.

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On small energy stabilization in the NLKG with a trapping potential

Cited by 11 publications

References 39 publications

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

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

On stabilization of small solutions in the nonlinear Dirac equation with a trapping potential

Solutions with Compact Time Spectrum to Nonlinear Klein–Gordon and Schrödinger Equations and the Titchmarsh Theorem for Partial Convolution

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