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

Cheng, Xing; Guo, Zihua; Yang, Kailong; Zhao, Liu

doi:10.4171/rmi/1155

Cited by 27 publications

(45 citation statements)

References 28 publications

(48 reference statements)

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“…We should point out that their study on "wave guide" manifolds seems to be of particular interest, especially in nonlinear optics of telecommunications [16,30,33]. In addition, the infinite dimensional vector-valued resonant nonlinear Schrödinger systems have also appeared in the study of the asymptotic behavior of the defocusing nonlinear Schrödinger equation on "wave guide" manifolds which are partially compact like R 2 × T in [7] and R × T 2 in [22]. In fact, from Hani and Pausader [22] and Cheng, Guo, Yang and Zhao [7], the infinite dimensional vector-valued resonant system (1.1) is derived during the construction of profile decomposition, which is an important step to get scattering of the defocusing nonlinear Schrödinger equation on corresponding "wave guide" manifolds.…”

Section: Introductionmentioning

confidence: 99%

Global Well-Posedness and Scattering for Mass-Critical, Defocusing, Infinite Dimensional Vector-Valued Resonant Nonlinear Schrödinger System

Yang¹,

Zhao²

2018

SIAM J. Math. Anal.

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In this article, we consider the infinite dimensional vector-valued resonant nonlinear Schrödinger system, which arises from the study of the asymptotic behavior of the defocusing nonlinear Schrödinger equation on "wave guide" manifolds like R 2 × T in [7]. We show global well-posedness and scattering for this system by long time Strichartz estimates and frequency localized interaction Morawetz estimates. As a by-product, our results make the arguments of scattering theory in [7] closed as crucial ingredients for compactness of the critical elements.The other is the frequency localized interaction Morawetz estimate Theorem 1.3 (Frequency localized interaction Morawetz estimate). Suppose ⃗ u(t, x) is a minimal mass blowup solution to (2.1) on [0, T ] with ∫ T 0 N(t) 3 dt = K. Then (1.4) j∈Z ∇ 1 2 P ≤ 10K ǫ 1 u j (t, x) 2 2 L 2 t,x ([0,T ]×R 2 ) ≲ o(K), where o(K) is a quantity such that o(K) K → 0 as K → ∞.

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Section: Introductionmentioning

confidence: 99%

Global Well-Posedness and Scattering for Mass-Critical, Defocusing, Infinite Dimensional Vector-Valued Resonant Nonlinear Schrödinger System

Yang¹,

Zhao²

2018

SIAM J. Math. Anal.

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“…Next, we establish the inverse Strichartz inequality along the L 2 -track by using the arguments from the proof of Lemma 3.1 and from [28,14]. For each j ∈ Z, define C j by…”

Section: Double Track Profile Decompositionmentioning

confidence: 99%

“…14),(4.18) and(4.22) then imply that d 2 dλ 2 H(T λ u(t)) ≤ 0 for all λ ∈ [1, λ * ].Finally, combining with (4.20), the fact that K(T λ * u(t)) = 0 and Taylor expansion we infer that d δ(d − 2) (u(t)) ≥ (λ * − 1) d dλ H(T λ u(t)) λ=1 ≥ H(T λ * u(t)) − H(u(t)) ≥ m M(u(0)) − H(u(0)). (4.23) This together with (4.15) yields (4.13).…”

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confidence: 95%

On sharp scattering threshold for the mass-energy double critical NLS via double track profile decomposition

Luo¹

2021

Preprint

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The present paper is concerned with the large data scattering problem for the mass-energy double critical NLSIn the defocusing-defocusing regime, Tao, Visan and Zhang show that the unique solution of DCNLS is global and scattering in time for arbitrary initial data in H 1 (R d ). This does not hold when at least one of the nonlinearities is focusing, due to the possible formation of blow-up and soliton solutions. However, precise thresholds for a solution of DCNLS being scattering were open in all the remaining regimes. Following the classical concentration compactness principle, we impose sharp scattering thresholds in terms of ground states for DCNLS in all the remaining regimes. The new challenge arises from the fact that the remainders of the standard L 2 -or Ḣ1profile decomposition fail to have asymptotically vanishing diagonal L 2 -and Ḣ1 -Strichartz norms simultaneously. To overcome this difficulty, we construct a double track profile decomposition which is capable to capture the low, medium and high frequency bubbles within a single profile decomposition and possesses remainders that are asymptotically small in both of the diagonal L 2 -and Ḣ1 -Strichartz spaces.

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“…The nonlinear Schrödinger equations on product spaces such as (1.2) have been intensively studied. See for example [3,4,8,9,10,11,18,19,22] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Pitchfork bifurcation at line solitons for nonlinear Schrödinger equations on the product space $\mathbb{R} \times \mathbb{T}$

Akahori¹,

Bahri²,

Ibrahim³

et al. 2021

Preprint

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We study a bifurcation problem for a stationary nonlinear Schrödinger equations with a pure power nonlinearity of exponent p > 1 on the product space (x, y) ∈ R×T.The equation has an explicit solution, called line soliton, which is independent of the transverse direction y. We construct a bifurcation branch from the line soliton.Such a result has already been known when p ≥ 2, and our main result extends the construction of the branch to the full range of nonlinear exponents p > 1. Since our condition on the nonlinearity is weak, we need to elaborate more when we apply the Crandall-Rabinowitz Theorem [5], taking full advantage of the crucial qualitative properties of the line soliton.

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

Cited by 27 publications

References 28 publications

Global Well-Posedness and Scattering for Mass-Critical, Defocusing, Infinite Dimensional Vector-Valued Resonant Nonlinear Schrödinger System

Global Well-Posedness and Scattering for Mass-Critical, Defocusing, Infinite Dimensional Vector-Valued Resonant Nonlinear Schrödinger System

On sharp scattering threshold for the mass-energy double critical NLS via double track profile decomposition

Pitchfork bifurcation at line solitons for nonlinear Schrödinger equations on the product space $\mathbb{R} \times \mathbb{T}$

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