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S. Cuccagna, E. Kirr, and D. Pelinovsky

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Parametric resonance of ground states in the nonlinear Schrodinger equation

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J. Diff. Eqs. 220, 85-120 (2006)

**Abstract:**

We study the global existence and long-time behavior of solutions of
the initial-value problem for the cubic nonlinear Schrodinger
equation with a linear attractive localized potential and a
time-dependent nonlinearity coefficient. For small initial data, we
show under some non-degeneracy assumptions that the solution
approaches the profile of the ground state and decays in time like
t^{-1/4}. The decay is due to resonant coupling between the ground
state and the radiation field induced by the time-dependent
nonlinearity coefficient.

**Keywords**:

SPECTRAL THEORY, NONLINEAR SCHRODINGER EQUATION, BOUND STATES,
PARAMETRIC RESONANCE, FERMI GOLDEN RULE, RADIATIVE DECAY