Abstract
We propose an approach to constructing semiclassical solutions for the generalized multidimensional Gross–Pitaevskii equation with a nonlocal interaction term. The key property of the solutions is that they are concentrated on a one-dimensional manifold (curve) that evolves over time. The approach reduces the Cauchy problem for the nonlocal Gross–Pitaevskii equation to a similar problem for the associated linear equation. The geometric properties of the resulting solutions are related to Maslov’s complex germ, and the symmetry operators of the associated linear equation lead to the approximation of the symmetry operators for the nonlocal Gross–Pitaevskii equation.
Highlights
The development of new mathematical methods for studying Bose–Einstein condensates (BECs) with complex geometric and topological properties is motivated by the progress in experimental physics
The construction of asymptotic solutions localized on incomplete Lagrangian manifolds to the nonlocal Gross–Pitaevskii equation (GPE) contributes to the theory of integro-differential equations of mathematical physics, but may have prospects in physical applications, e.g., for describing extended quantum objects like BECs in magnetic traps of complex geometries
We solve this problem starting from the Maslov–WKB semiclassical approximation applied to the nonlocal GPE
Summary
The development of new mathematical methods for studying Bose–Einstein condensates (BECs) with complex geometric and topological properties is motivated by the progress in experimental physics. Analytical approaches for the nonlocal GPE were studied in [23], particular solutions were found in [24], and the collapse problem of the localized waves described by (2) was discussed in [25]. The rigorous theory of the semiclassical quantization of nonintegrable Hamilton systems for linear h−1 -(pseudo)differential operators is based on the Maslov complex germ method [33,34,35] In this theory, the problem of the construction of semiclassical asymptotics is reduced to the construction of geometric objects in a 2n-dimensional phase space (family of Lagrangian manifolds Λk with a complex germ r n ). We partially generalize these results to a one-dimensional manifold Λ1 and propose a new method for constructing the Cauchy problem’s approximate solutions concentrated on a curve for a multidimensional nonlocal GPE.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
