The threshold for the focusing Gross–Pitaevskii equation with trapped dipolar quantum gases

Abstract

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.

Blow up threshold for the Gross–Pitaevskii system with combined nonlocal nonlinearities

“…Our present paper is directly motivated by [3,12,10,11]. We generalize the above results to the nonlocal Gross-Pitaevskii system (1) with K(x) written as in (3).…”

“…In physics, the answers to questions under what conditions, will the condensate exist for all time, or under what conditions, will the condensate become unstable to collapse are pursued strongly [3][4][5]7,11,12,10,13,15,20,23].…”

“…Also, the unstable regime is obtained, i.e., when λ 1 < 4 3 λ 2 , and 3E ≤ xu 0 2 2 , the solution of (2) blows up in finite time. To deal with the other case that for λ 1 < 4 3 λ 2 , and 3E > xu 0 2 2 , Ma and Cao [11], Ma and Wang [12] considered Eq. (2), and obtained the blow up threshold for the solution.…”

Blow up threshold for the Gross‐Pitaevskii system with trapped dipolar quantum gases

“…In physics, the answers to questions that under what conditions, will the condensate exist for all time; while under what conditions, will the condensate become unstable to collapse are pursued strongly .…”

“…Also, the unstable regime is obtained, i.e., when , and , the solution of blows up in finite time. To deal with the other case that for , and , Ma and Cao , Ma and Wang considered Eq. , and obtained the blow up threshold for the solution.…”