Abstract
The study of nonlinear Schrödinger systems with quadratic interactions has attracted much attention in the recent years. In this paper, we summarize time decay estimates of small solutions to the systems under the mass resonance condition in 2-dimensional space. We show the existence of wave operators and modified wave operators of the systems under some mass conditions in n-dimensional space, where n ≥ 2. The existence of scattering operators and finite time blow-up of the solutions for the systems in higher space dimensions is also shown.
Highlights
In this paper we survey recent progress on asymptotic behavior of solutions to nonlinear Schrodinger system, i∂tVj + 1 2mj ΔVj = Fj (V1, . , Vl) Gj (Vj), t ∈ R, x ∈ Rn, (1)Vj (0, x) = φj, x ∈ Rn, based on papers [1,2,3,4,5,6,7,8,9], where 1 ≤ j ≤ l, Vj is the complex conjugate of Vj, mj is a mass of particle, and nonlinearities have the form
If we let uj = e−itmjc2 Vj in (5), by the condition (6) we find that Vj satisfies
The main task is to show that remainder terms are estimated from above by O(t−1−ε) which is integrable in time
Summary
In this paper we survey recent progress on asymptotic behavior of solutions to nonlinear Schrodinger system, i∂tVj. The quadratic nonlinearities of nonlinear Schrodinger systems in two space dimensions are interesting mathematical problems since they are regarded as the borderline between short range and long range interactions In this case, asymptotic behavior of solutions to nonlinear systems is different from that to linear systems under mass resonance conditions and is the same as that to linear systems under mass nonresonance conditions. N} is an important tool to study time decay of solutions to nonlinear Schrodinger equations satisfying the gauge invariant condition (6) since it acts as a differential operator. In the last section we consider the related and open problems
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