Quasi-energy spectral series for a nonlocal Gross-Pitaevskii equation

Abstract

For a nonlocal Gross-Pitaevskii operator, quasi-energy spectral series have been constructed accurate to O( 3/2 ).These series correspond to the closed phase trajectories of the Hamilton-Ehrenfest system, which are stable in a linear approximation. The corresponding monodromy operator has been constructed accurate toin the class of trajectory-coherent functions.

The Gross–Pitaevskii Equation with a Nonlocal Interaction in a Semiclassical Approximation on a Curve

“…In this section, we seek a solution concentrated on a curve Λ 1 t for the Gross-Pitaevskii equation in the sense of (7). To this end, we involve the results of [26,28,[42][43][44], where asymptotic solutions to the nonlocal GPE were constructed in the class P t h of the trajectory-concentrated functions (15) accurate to O(h 3/2 ) using an associated linear equation of the Schrödinger type.…”

“…The moments of the trajectory-concentrated functions can be obtained from an auxillary system of differential equations which are semiclassical approximations to the Ehrenfest equation (see [26][27][28][42][43][44]). In the class J τ h , the expectations are given by (58).…”

“…The asymptotics in the class of functions semiclassically concentrated on zero-dimensional manifolds Λ 0 were constructed in [26][27][28][42][43][44] (see also [45]).…”