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arXiv · 2201.02690

The 3D nonlinear Schr\"odinger equation with a constant magnetic field revisited

Van Duong Dinh
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Abstract

In this paper, we revisit the Cauchy problem for the three dimensional nonlinear Schr\"odinger equation with a constant magnetic field. We first establish sufficient conditions that ensure the existence of global in time and finite time blow-up solutions. In particular, we derive sharp thresholds for global existence versus blow-up for the equation with mass-critical and mass-supercritical nonlinearities. We next prove the existence and orbital stability of normalized standing waves which extend the previous known results to the mass-critical and mass-supercritical cases. To show the existence of normalized solitary waves, we present a new approach that avoids the celebrated concentration-compactness principle. Finally, we study the existence and strong instability of ground state standing waves which greatly improve the previous literature.

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Van Duong Dinh. 2022-01-07. The 3D nonlinear Schr\"odinger equation with a constant magnetic field revisited. https://arxiv.org/abs/2201.02690

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