A Note on Inhomogeneous Coupled Schrödinger Equations

Ghanmi, Radhia; Hezzi, H.; Saanouni, Tarek

doi:10.1007/s00023-020-00942-0

Cited by 7 publications

(4 citation statements)

References 32 publications

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“…The local well-posedness of the above problem (1.1) in the energy space for 2 ≤ p < p * was proved in [5] under the assumptions denoted for simplicity (N, α, b) satisfies (C) if, 0 < α < N and max{−(N + α), −4(1 + α N ), N − 8 − α} < 2b < 0 and [N ≥ 5 or 3 ≤ N ≤ 4 and 2α + 4b + N > 0]. Moreover, the global existence versus finite time blow-up of energy solutions under the ground state threshold was obtained in [14].…”

Section: Background and Main Resultsmentioning

confidence: 99%

“…Remarks 2.5. Note that 1. the global existence of solutions under the assumption (2.2) was proved in [5]; 2. the proof follows a new approach due to [4] and avoids the concentrationcompactness method introduced by [10]; 3. the main novelty here is the removal of the spherically symmetric of the data; 4. one exploits the decay of the inhomogeneous term | • | b of the source term, instead of the spherically symmetric assumption; 5. the condition N ≥ 5 is used in the scattering criteria; 6. it seems that the scattering for non-radial data in lower dimensions still remains open; 7. in the limiting case b = 0, it seems that the scattering for non-radial data is still open.…”

Section: Background and Main Resultsmentioning

confidence: 99%

“…See also [12], for the well-posedness in H s , 0 < s ≤ 2 and [2] for existence of global and non-global solutions in Ḣsc ∩ Ḣ2 . The case of a non-local source term was considered by the author in [5]. The scattering of global spherically symmetric solutions under the ground state threshold was established recently by the author [14].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Scattering for Radial Defocusing Inhomogeneous Bi-Harmonic SchrÖDinger Equations

Saanouni

2021

Potential Anal

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Section: Background and Main Resultsmentioning

confidence: 99%

Section: Background and Main Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Scattering for Radial Defocusing Inhomogeneous Bi-Harmonic SchrÖDinger Equations

Saanouni

2021

Potential Anal

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“…Note that the case of spatial growing nonlinearity, that is,

b&amp;lt;0

, has been recently investigated in Dinh et al [25]. See also previous studies [26–30].…”

Section: Introductionmentioning

confidence: 99%

Long‐time dynamics for the radial focusing fractional INLS

Majdoub

Saanouni

2023

Math Methods in App Sciences

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We consider the following fractional NLS with focusing inhomogeneous power‐type nonlinearity: where , , , and . We prove the ground state threshold of global existence and scattering versus finite time blowup of energy solutions in the inter‐critical regime with spherically symmetric initial data. The scattering is proved by the new approach of Dodson–Murphy. This method is based on Tao's scattering criteria and Morawetz estimates. We describe the threshold using some non‐conserved quantities in the spirit of the recent paper by Dinh. The radial assumption avoids a loss of regularity in Strichartz estimates. The challenge here is to overcome two main difficulties. The first one is the presence of a non‐local fractional Laplacian operator. The second one is the presence of a singular weight in the nonlinearity. The greater part of this paper is devoted to the scattering of global solutions in . The Lorentz spaces and the Strichartz estimates play crucial roles in our approach.

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Coupled inhomogeneous nonlinear Schrödinger system with potential in three space diemensions

Saanouni,

Ghanmi

2024

Math Methods in App Sciences

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This work studies the inhomogeneous Schrödinger coupled system with potential The wave function components read for . The inhomogeneous singular term exponent is . The source term is energy sub‐critical: . Moreover, in order to avoid a singularity of the term , one assumes that . The linear Schrödinger operator reads , where is a potential satisfying some assumptions which imply that the dispersive and Strichartz estimates hold. The purpose is two‐fold. First, we develop a local well‐posedness theory in the energy space . Second, we present a dichotomy of global existence and scattering versus blow‐up of energy solutions under the ground‐state threshold in the inter‐critical focusing regime. The scattering is obtained by using the new approach of Dodson–Murphy which is based on Tao's scattering criteria and Morawetz estimates. The novelty here is the presence of the potential . The challenge is to deal with three technical problems: a coupled nonlinearity, an inhomogeneous singular term , and the presence of the potential . This note naturally extends previous works by the authors about the above problem for .

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A Note on Inhomogeneous Coupled Schrödinger Equations

Cited by 7 publications

References 32 publications

Scattering for Radial Defocusing Inhomogeneous Bi-Harmonic SchrÖDinger Equations

Scattering for Radial Defocusing Inhomogeneous Bi-Harmonic SchrÖDinger Equations

Long‐time dynamics for the radial focusing fractional INLS

Coupled inhomogeneous nonlinear Schrödinger system with potential in three space diemensions

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