Semiclassical spectral series for the two-component Hartree-type equation

Abstract

Semiclassical spectral series for the two-component nonlinear Hartree-type operator corresponding to the rest point of the phase curve are constructed on the basis of the Maslov complex germ theory.

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

“…System has applications in many physical problems, such as nonlinear optics for a description of pulse propagation . Moreover, it plays a fundamental role in the theory of Bose‐Einstein condensate . In the BEC theory, it describes the Bose‐Einstein condensate of several types of trapped bosons at ultralow temperature, and the solution denotes the quantum states of the j ‐th component of the condensate.…”

“…In the BEC theory, it describes the Bose‐Einstein condensate of several types of trapped bosons at ultralow temperature, and the solution denotes the quantum states of the j ‐th component of the condensate. The N ‐component condensate is a mixture of N systems of different atoms or an ensemble of atoms of one species with different spin states . The external harmonic potential is the same for each condensate components, which describes the trap formed by laser or the electromagnetic.…”

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

“…System (1) plays a fundamental role in the theory of Bose-Einstein condensate [2] and also has applications in nonlinear optics [9]. In the BEC theory, it describes the Bose-Einstein condensate of several types of trapped bosons at ultralow temperature, and the solution u j denotes the quantum states of the j-th component of the condensate.…”

“…In the BEC theory, it describes the Bose-Einstein condensate of several types of trapped bosons at ultralow temperature, and the solution u j denotes the quantum states of the j-th component of the condensate. The N -component condensate is a mixture of N systems of different atoms or an ensemble of atoms of one species with different spin states [2]. The interaction among components of the system is considered by the combined nonlocal nonlinear terms.…”

“…Using the Banach fixed point theorem and the Strichartz estimates [6,8,21], they obtained the existence and uniqueness result for the Cauchy problem (2). Among other things, they obtained the stable regime, i.e., when λ 1 ≥ 4π 3 λ 2 ≥ 0, or λ 1 ≥ − 8 3 πλ 2 ≥ 0, the solution is global in time.…”