For a broad class of quadratic-type Schrödinger systems in dimensions 1 through 6, the paper proves that ground states of the elliptic system govern the global-existence/blow-up dichotomy and the stability of standing waves.
On the dynamics of a quadratic Schr\"odinger system in dimension $n=5$
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abstract
In this work we give a sharp criterion for the global well-posedness, in the energy space, for a system of nonlinear Schr\"odinger equations with quadratic interaction in dimension $n=5$. The criterion is given in terms of the charge and energy of the ground states associated with the system, which are obtained by minimizing a Weinstein-type functional. The main result is then obtained in view of a sharp Gagliardo-Nirenberg-type inequality.
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2019 1verdicts
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On a system of Schr\"odinger equations with general quadratic-type nonlinearities
For a broad class of quadratic-type Schrödinger systems in dimensions 1 through 6, the paper proves that ground states of the elliptic system govern the global-existence/blow-up dichotomy and the stability of standing waves.