Abstract

We study the asymptotic dynamics for solutions to a system of nonlinear Schrödinger equations with cubic interactions, arising in nonlinear optics. We provide sharp threshold criteria leading to global well-posedness and scattering of solutions, as well as formation of singularities in finite time for (anisotropic) symmetric initial data. The free asymptotic results are proved by means of Morawetz and interaction Morawetz estimates. The blow-up results are shown by combining variational analysis and an ODE argument, which overcomes the unavailability of the convexity argument based on virial-type identities.

中文翻译：

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在具有 χ3 非线性响应的光学材料中产生的 NLS 系统的散射和爆炸的尖锐条件

摘要

我们研究了非线性薛定谔方程组的解的渐近动力学，该方程组具有三次相互作用，出现在非线性光学中。我们提供了尖锐的阈值标准，导致解决方案的全局适定性和散射，以及在有限时间内为（各向异性）对称初始数据形成奇点。自由渐近结果通过 Morawetz 和交互 Morawetz 估计证明。通过结合变分分析和 ODE 论证来显示膨胀结果，这克服了基于维里型恒等式的凸性论证的不可用性。