Global well-posedness and scattering for mass-critical, defocusing, infinite dimensional vector-valued resonant nonlinear Schrödinger system

Yang, Kailong; Zhao, Liu

doi:10.48550/arxiv.1704.08976

Cited by 4 publications

(11 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Title Scattering for the cubic NLS on R 2 × T Theorem 1.2 (Scattering of the cubic resonant Schrödinger system, [26]). Let E > 0, for any initial data…”

mentioning

confidence: 99%

On scattering for the cubic defocusing nonlinear Schrödinger equation on the waveguide $\mathbb R^2 \times \mathbb T$

et al. 2020

Self Cite

View full text Add to dashboard Cite

In this article, we will show the global wellposedness and scattering of the cubic defocusing nonlinear Schrödinger equation on waveguide R 2 ×T in H 1 . We first establish the linear profile decomposition in H 1 (R 2 × T) motivated by the linear profile decomposition of the mass-critical Schrödinger equation in L 2 (R 2 ). Then by using the solution of the infinite dimensional vector-valued resonant nonlinear Schrödinger system to approximate the nonlinear profile, we can prove scattering in H 1 by using the concentrationcompactness/rigidity method.

show abstract

“…Title Scattering for the cubic NLS on R 2 × T Theorem 1.2 (Scattering of the cubic resonant Schrödinger system, [26]). Let E > 0, for any initial data…”

mentioning

confidence: 99%

On scattering for the cubic defocusing nonlinear Schrödinger equation on the waveguide $\mathbb R^2 \times \mathbb T$

et al. 2020

Self Cite

View full text Add to dashboard Cite

show abstract

“…This abstract linear profile decomposition is a infinite vector version of linear profile decomposition of the mass-critical Schrödinger equations, and the proof relies on the proof of the linear profile decomposition of the mass-critical Schrödinger equations, especially the bilinear Strichartz estimate of the Schrödinger equation on R d . This kind of linear profile decomposition is of independent interest, it can be used in the proof of the scattering of the vector-valued nonlinear Schrödinger system [55], and also can be used in the study of well-posedness and the long time behavior of the nonlinear dispersive equations such as…”

Section: Introductionmentioning

confidence: 99%

“…Compared to the cubic resonant system problem in [55], the quintic resonant relation is more complicated and we need to take care the nonlinearity more delicately. Remarkably, first we reduce the scattering norm from L 6…”

Section: Introductionmentioning

confidence: 99%

“…x,t h 1 to L 6 x,t h β ( 3 8 < β < 1) by establishing nonlinear estimate to make it possible to use profile decomposition to obtain the almost periodic solution. For the 2d cubic resonant system problem (see [55]), since the resonance is less complicated, the authors can even reduce the the scattering norm from L 4 t,x h 1 to L 4 t,x l 2 . Since our goal is to show the scattering norm of the solution is finite and a smaller quantity is of course more likely to be finite, it is easier for us to reduce the scattering norm to a smaller one.…”

Section: Introductionmentioning

confidence: 99%

“…, and ξ(t 1 ) − ξ(t 2 ) ≲ N(J), where N(J) ∶= sup t∈J N(t). In addition, [7,13,14,32,37,38,53,55] for the application of the long time Stricharz estimate for other dispersive problems. Similar as the mass-critical nonlinear Schrödinger equations, the long time Stricharz estimate mainly helps us in two aspects, i.e.…”

mentioning

confidence: 99%

See 2 more Smart Citations

On scattering for the defocusing quintic nonlinear Schrödinger equation on the two-dimensional cylinder

Cheng¹,

Guo²,

Zhao³

2018

Preprint

View full text Add to dashboard Cite

In this article, we prove the scattering for the quintic defocusing nonlinear Schrödinger equation on cylinder R × T in H 1 . We establish an abstract linear profile decomposition in L 2x h α , 0 < α ≤ 1, motivated by the linear profile decomposition of the mass-critical Schrödinger equation in L 2 (R d ), d ≥ 1. Then by using the solution of the one-discrete-component quintic resonant nonlinear Schrödinger system, whose scattering can be proved by using the techniques in 1d mass critical NLS problem by B. Dodson, to approximate the nonlinear profile, we can prove scattering in H 1 by using the concentration-compactness/rigidity method. As a byproduct of our proof of the scattering of the one-discrete-component quintic resonant nonlinear Schrödinger system, we also prove the conjecture of the global well-posedness and scattering of the two-discrete-component quintic resonant nonlinear Schrödinger system made by Z. Hani and B. Pausader [Comm. Pure Appl. Math. 67 (2014)].

show abstract

Global well-posedness and scattering for the defocusing cubic Schrödinger equation on waveguide $\mathbb{R}^2$ $\times$ $\mathbb{T}^2$

Zhao

2017

Preprint

View full text Add to dashboard Cite

In this paper, we consider the problem of large data scattering for the defocusing cubic nonlinear Schrödinger equation on R 2 × T 2 . This equation is critical both at the level of energy and mass. The key ingredients are global-in-time Stricharz estimate, resonant system approximation, profile decomposition and energy induction method. Assuming the large data scattering for the 2d cubic resonant system, we prove the large data scattering for this problem. This problem is the cubic analogue of [13].

show abstract

Global well-posedness and scattering for mass-critical, defocusing, infinite dimensional vector-valued resonant nonlinear Schrödinger system

Cited by 4 publications

References 12 publications

On scattering for the cubic defocusing nonlinear Schrödinger equation on the waveguide $\mathbb R^2 \times \mathbb T$

On scattering for the cubic defocusing nonlinear Schrödinger equation on the waveguide $\mathbb R^2 \times \mathbb T$

On scattering for the defocusing quintic nonlinear Schrödinger equation on the two-dimensional cylinder

Global well-posedness and scattering for the defocusing cubic Schrödinger equation on waveguide $\mathbb{R}^2$ $\times$ $\mathbb{T}^2$

Contact Info

Product

Resources

About