Semiclassical Spectral Series Localized on a Curve for the Gross–Pitaevskii Equation with a Nonlocal Interaction

Abstract

We propose the approach to constructing semiclassical spectral series for the generalized multidimensional stationary Gross–Pitaevskii equation with a nonlocal interaction term. The eigenvalues and eigenfunctions semiclassically concentrated on a curve are obtained. The curve is described by the dynamic system of moments of solutions to the nonlocal Gross–Pitaevskii equation. We solve the eigenvalue problem for the nonlocal stationary Gross–Pitaevskii equation basing on the semiclassical asymptotics found for the Cauchy problem of the parametric family of linear equations associated with the time-dependent Gross–Pitaevskii equation in the space of extended dimension. The approach proposed uses symmetries of equations in the space of extended dimension.

Semiclassical Approach to the Nonlocal Kinetic Model of Metal Vapor Active Media

“…i.e., g(0, C ϕ ) = g ϕ . We consider the solution of Equation (33) in specific examples, leaving aside the general algebraic problem of solvability of this equation. Note also that C ϕ is a vector functional of ϕ.…”

“…Replace the arbitrary constants C in Equation ( 40) or (42) by the constants C ϕ determined by the algebraic condition (33). Considering (34), (37), and (45), we can see that Equation (40) or (42) transforms into Equation (39) accurate to O(D 3/2 ).…”

“…The method of semiclassical asymptotics was applied in [28][29][30] to a nonlocal generalization of the Fisher-Kolmogorov-Petrovskii-Piskunov equation known in the theory of biological populations, and also in [31][32][33] for the nonlocal Gross-Pitaevskii equation, which is widely used in the theory of Bose-Einstein condensates. The approach proposed here for the kinetic equation of plasma ionization essentially involves the results of [28][29][30].…”

A Semiclassical Approach to the Nonlocal Nonlinear Schrödinger Equation with a Non-Hermitian Term

“…The first identity in (38) follows from ( 21) and (31), while the second one follows from ( 21) and ( 16). Thus, (37) can be written as follows:…”

“…Some modern semiclassical approaches based on ideas of the Maslov method [33] were also applied to some nonlinear problems (see, e.g., [34,35]). In [36][37][38], the formalism of semiclassical asymptotics for a generalized nonlocal GPE in a special class of trajectory concentrated functions is developed that corresponds to closed quantum systems. In [39][40][41], this formalism was applied to kinetic reaction-diffusion equations that correspond to open classical systems.…”