We study the propagation of wave packets for nonlinear nonlocal Schrödinger equations in the semi-classical limit. When the kernel is smooth, we construct approximate solutions for the wave functions in subcritical, critical and supercritical cases (in terms of the size of the initial data). The validity of the approximation is proved up to Ehrenfest time. For homogeneous kernels, we establish similar results in subcritical and critical cases. Nonlinear superposition principle for two nonlinear wave packets is also considered.

This paper concerns the threshold of global existence and finite time blow up of solutions to the time-dependent focusing Gross-Pitaevskii equation describing the BoseEinstein condensation of trapped dipolar quantum gases. Via a construction of new crossconstrained invariant sets, it is shown that either the corresponding solution globally exists or blows up in finite time according to some appropriate assumptions about the initial datum.

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We study the standing wave solutions for nonlinear Hartree equations. Of special interest to us is the existence and orbital stability of the standing wave solutions for the Hartree equation in the presence of confining potential. We establish the existence of two different notions of ground states via variation methods and orbital stability results for the corresponding standing wave solutions for focusing Hartree equations. The existence of the standing waves for defocusing Hartree equation is also considered.

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