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.
Cazenave, Semilinear Schrödinger Equations , American Mathematical Society, 2003
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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.