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

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.

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

“…Here,ẑ = (ˆ p, x), x ∈ R n ,ˆ p = −ih∂ x , and V(ẑ) is a linear partial differential operator depending onẑ with a symbol V(z) = V( p, x) smoothly depending on p and x ( p ∈ R n ). Our approach to the problem (7) relies on the method proposed in [25] for solution of the Cauchy problem in the semiclassical approximation for the non-stationary nonlocal GPE:…”

“…The core idea of the approach proposed is as follows. The method of [25] provides a practical possibility to select from the set of asymptotical solutions Ψ( x, t) to the Cauchy problem (8), found for various initial functions ψ( x), those Ψ( x, t) that have the form…”

“…For the functions Ψ semiclassically concentrated on the curve Λ 1 t , the dynamics of the curve in the limitsh → 0 is determined by the (1, 1)-type Hamilton-Ehrenfest system (the term (1, 1) implies that the parameter s is a one-dimensional variable and the vector Z(s, t) corresponds to the first order moments of the function Ψ( x, t), see [25] for details):…”

“…We apply the well-known WKB-Maslov theory of semiclassical approximation [22][23][24] to the problem under consideration. The approach to the eigenvalue problem (4) and ( 5) is based on [25], where the leading term of semiclassical asymptotics for the Cauchy problem has been constructed explicitly in a class of functions localized in a neighborhood of a curve in the phase space of a dynamic system of moments of the nonlocal GPE solution termed the Hamilton-Ehrenfest system. The last one for the n-dimensional manifolds in the phase space was studied in [26][27][28].…”

“…The last one for the n-dimensional manifolds in the phase space was studied in [26][27][28]. As opposed to the linear case (κ = 0 in (4)), the variance matrix of the solutions constructed in [25] may be bounded due to the nonlinearity. Such solutions of the Cauchy problems are termed the semiclassical soliton-type solutions [29], and we use exactly such particular solutions for the construction of the spectral series of the nonlocal GPE.…”