We study the global existence and long-time behavior of solutions of the initial-value problem for the cubic nonlinear Schr\" odinger equation with an 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.
Parametric resonance of ground states in the nonlinear Schr"odinger equation
CUCCAGNA, SCIPIO;
2006-01-01
Abstract
We study the global existence and long-time behavior of solutions of the initial-value problem for the cubic nonlinear Schr\" odinger equation with an 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.File in questo prodotto:
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