Global well-posedness, scattering, and blowup for nonlinear coupled Schrödinger equations in ℝ<sup>3</sup>

Xu, Yushun

doi:10.1080/00036811.2015.1015523

Cited by 11 publications

(9 citation statements)

References 17 publications

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“…Intensive work has been done in the last few years about coupled Schrödinger systems [17,28,16,22]. These works have been mainly on 2-systems or with small couplings.…”

Section: For 1 < P < N +2mentioning

confidence: 99%

On coupled nonlinear Schrodinger equations with harmonic potential

Saanouni¹

2015

Preprint

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“…Intensive work has been done in the last few years about coupled Schrödinger systems [17,28,16,22]. These works have been mainly on 2-systems or with small couplings.…”

Section: For 1 < P < N +2mentioning

confidence: 99%

On coupled nonlinear Schrodinger equations with harmonic potential

Saanouni¹

2015

Preprint

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“…Although there are some results on the scattering theory for a system of Schrödinger equations(see [12,31,93]), some new difficulties arising in the study of the system of Schrödinger equations with more general nonlinearities. Besides the local wellposedness results, differing to the scalar case, some Hamiltonian may be difficult to get.…”

Section: Introductionmentioning

confidence: 99%

Critical exponents line, scattering thoeries for a weighted gradient system of semilinear Schrodinger equations

Song

2021

Preprint

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In this paper, we consider the following Cauchy problem of a weighted(or essential) gradient system of semilinear Schrödinger equationsUnder certain assumptions, we establish the local wellposedness of the H 1 × H 1 -solution, Ḣ1 × Ḣ1 -solution and H s × H s -solution of the system with different types of initial data.If f (|u| 2 , |v| 2 )u = λ|u| α |v| β+2 u and g(|u| 2 , |v| 2 )v = µ|u| α+2 |v| β v with λ > 0, µ > 0, α ≥ 0 and β ≥ 0, it is surprised that there exists a critical exponents line α + β = 2 when d = 3 in the following sense: The system always has a unique bounded H 1 × H 1 -solution for any initial data (u0, v0) ∈ H 1 (R 3 ) × H 1 (R 3 ) if α + β ≤ 2, yet we can find some initial data (u0, v0) ∈ H 1 (R 3 ) × H 1 (R 3 ) such that it doesn't possess the global H 1 × H 1 -solution if α + β > 2 and α = β. While when d = 4, we call (α, β) = (0, 0) is the critical exponents point in the following sense: The system always has a unique bounded H 1 ×H 1 -solution for any initial data (u0, v0)Moreover, we establish the H 1 × H 1 and Σ × Σ scattering theories for the solution if α + β < 2(i.e., (α, β) is below the critical exponents line α + β = 2) when d = 3, Σ × Σ scattering theory for the solution if (α, β) is on the critical exponents line α + β = 2 excluding the endpoints when d = 3, Ḣ1 × Ḣ1 scattering theory for the solution if (α, β) is the endpoint of the critical exponents line α + β = 2 when d = 3 and (α, β) = (0, 0) when d = 4. We also establish Ḣsc × Ḣsc scattering theories for the corresponding solutions if (α, β) is above the critical exponents line α + β = 2 when d = 3 and (α, β) > (0, 0) when d ≥ 4. Here sc = d 2 − 2 α+β+2 .

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“…Intensive work has been done in the last few years about coupled Schrödinger systems [16,27,15]. These works have been mainly on 2-systems or with small couplings.…”

Section: Introductionmentioning

confidence: 99%

A note on coupled focusing nonlinear Schrodinger equations

Saanouni¹

2015

Preprint

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Global well-posedness, scattering, and blowup for nonlinear coupled Schrödinger equations in ℝ³

Cited by 11 publications

References 17 publications

On coupled nonlinear Schrodinger equations with harmonic potential

On coupled nonlinear Schrodinger equations with harmonic potential

Critical exponents line, scattering thoeries for a weighted gradient system of semilinear Schrodinger equations

A note on coupled focusing nonlinear Schrodinger equations

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Global well-posedness, scattering, and blowup for nonlinear coupled Schrödinger equations in ℝ3

Cited by 11 publications

References 17 publications

On coupled nonlinear Schrodinger equations with harmonic potential

On coupled nonlinear Schrodinger equations with harmonic potential

Critical exponents line, scattering thoeries for a weighted gradient system of semilinear Schrodinger equations

A note on coupled focusing nonlinear Schrodinger equations

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Global well-posedness, scattering, and blowup for nonlinear coupled Schrödinger equations in ℝ³