Global Well-posedness for the Focusing Cubic NLS on the Product Space $\mathbb{R} \times \mathbb{T}^3$

Yu, Xueying; Yue, Haitian; Zhao, Zehua

doi:10.1137/20m1364953

Cited by 20 publications

(8 citation statements)

References 41 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Threshold assumptions are necessary and new ingredients are needed to handle this type of problems. See [37] for a global well-posedness result and [7,28] for two very recent scattering result. Moreover, see [14,24,27] for the Euclidean result.…”

Section: ḣ1mentioning

confidence: 99%

“…We refer to [8,13,25] for some important Euclidean results. Moreover, we refer to [6,5,18,20,21,22,23,26,33,34,37,39,40,41] with regard to the tori case and the waveguide case. We may roughly think of the waveguide case as the "intermediate point" between the Euclidean case and the tori case since the waveguide manifold is the product of the Euclidean space and the tori.…”

mentioning

confidence: 99%

See 1 more Smart Citation

On scattering asymptotics for the 2D cubic resonant system

Yang¹,

Zhao²

2022

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

Section: ḣ1mentioning

confidence: 99%

mentioning

confidence: 99%

On scattering asymptotics for the 2D cubic resonant system

Yang¹,

Zhao²

2022

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…and scattering of the defocusing nonlinear Schrödinger equations on the cylinder R 2 × T (See [9,10,51]), there are very few result on the long time behavior of the solutions of the focusing nonlinear Schrödinger equations on the cylinders. We refer to [49] for a result on the orbital stability in the focusing case, and global well-posedness result in [56].…”

Section: Introductionmentioning

confidence: 99%

Global well-posedness and scattering of the two dimensional cubic focusing nonlinear Schrödinger system

Xing¹,

Guo²,

Hwang³

et al. 2022

Preprint

View full text Add to dashboard Cite

In this article, we prove the global well-posedness and scattering of the cubic focusing infinite coupled nonlinear Schrödinger system on R 2 below the threshold in L 2x h 1 (R 2 × Z). We first establish the variational characterization of the ground state, and derive the threshold of the global well-posedness and scattering. Then we show the global well-posedness and scattering below the threshold by the concentration-compactness/rigidity method, where the almost periodic solution is excluded by adapting the argument in the proof of the mass-critical nonlinear Schrödinger equations by B. Dodson. As a byproduct of the scattering of the cubic focusing infinite coupled nonlinear Schödinger system, we obtain the scattering of the cubic focusing nonlinear Schrödinger equation on the small cylinder, this is the first large data scattering result of the focusing nonlinear Schrödinger equations on the cylinders. In the article, we also show the global well-posedness and scattering of the two dimensional N −coupled focusing cubic nonlinear Schrödinger system in L 2 (R 2 )N .

show abstract

“…Following the nowadays well-known concentration compactness arguments initiated by Kenig and Merle [35] and the so-called Black-Box-Theory, Ionescu, Pausader and Staffilani [30,31,32] showed that defocusing energy-critical NLS on T 3 , R × T 3 and on the hyperbolic space H 3 are always globally well-posed. By appealing to suitable variational arguments, Yu, Yue and Zhao [51,50] utilized the Black-Box-Theory to prove that solutions of the focusing energy-critical NLS on T 4 and R × T 3 lying below ground states are always globally well-posed.…”

Section: Introductionmentioning

confidence: 99%

Large data global well-posedness and scattering for the focusing cubic nonlinear Schrödinger equation on $\mathbb{R}^2\times\mathbb{T}$

Luo¹

2022

Preprint

View full text Add to dashboard Cite

We consider the focusing cubic nonlinear Schrödinger equation (NLS)Different from the 3D Euclidean case, the (3NLS) is mass-critical and non-scale-invariant on the waveguide manifold R 2 × T, hence the underlying analysis becomes more subtle and challenging. We formulate thresholds using the 2D Euclidean ground state of the focusing cubic NLS and show that solutions of (3NLS) lying below the thresholds are global and scattering in time. The proof relies on several new established Gagliardo-Nirenberg inequalities, whose best constants are formulated in term of the 2D Euclidean ground state. It is also worth noting the interesting fact that the thresholds for global well-posedness and scattering do not coincide. To the author's knowledge, this paper also gives the first large data scattering result for focusing NLS on product spaces.

show abstract

Global Well-posedness for the Focusing Cubic NLS on the Product Space $\mathbb{R} \times \mathbb{T}^3$

Cited by 20 publications

References 41 publications

On scattering asymptotics for the 2D cubic resonant system

On scattering asymptotics for the 2D cubic resonant system

Global well-posedness and scattering of the two dimensional cubic focusing nonlinear Schrödinger system

Large data global well-posedness and scattering for the focusing cubic nonlinear Schrödinger equation on $\mathbb{R}^2\times\mathbb{T}$

Contact Info

Product

Resources

About