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

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1289
Author(s):  
Anton E. Kulagin ◽  
Alexander V. Shapovalov ◽  
Andrey Y. Trifonov
Keyword(s):  
Cauchy Problem ◽  
Linear Equations ◽  
Time Dependent ◽  
Nonlocal Interaction ◽  
Pitaevskii Equation ◽  
Spectral Series ◽  
Semiclassical Asymptotics ◽  
Eigenvalues And Eigenfunctions ◽  
The Cauchy Problem ◽  
Interaction Term

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.

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

Symmetry ◽  
2020 ◽  
Vol 12 (2) ◽  
pp. 201
Author(s):  
Alexander V. Shapovalov ◽  
Anton E. Kulagin ◽  
Andrey Yu. Trifonov
Keyword(s):  
Linear Equation ◽  
Semiclassical Approximation ◽  
Nonlocal Interaction ◽  
Pitaevskii Equation ◽  
Dimensional Manifold ◽  
Complex Germ ◽  
One Dimensional ◽  
The Cauchy Problem ◽  
Symmetry Operators ◽  
Interaction Term

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.

scholarly journals Statistics of Gaussian packets on metric and decorated graphs

2014 ◽  
Vol 372 (2007) ◽  
pp. 20130145 ◽  
Author(s):  
V. L. Chernyshev ◽  
A. I. Shafarevich
Keyword(s):  
Cauchy Problem ◽  
Asymptotic Solution ◽  
Riemannian Manifolds ◽  
Initial Function ◽  
Time Dependent ◽  
Semiclassical Asymptotics ◽  
Subexponential Growth ◽  
Codimension One ◽  
The Cauchy Problem ◽  
Opposite Situation

We study a semiclassical asymptotics of the Cauchy problem for a time-dependent Schrödinger equation on metric and decorated graphs with a localized initial function. A decorated graph is a topological space obtained from a graph via replacing vertices with smooth Riemannian manifolds. The main term of an asymptotic solution at an arbitrary finite time is a sum of Gaussian packets and generalized Gaussian packets (localized near a certain set of codimension one). We study the number of packets as time tends to infinity. We prove that under certain assumptions this number grows in time as a polynomial and packets fill the graph uniformly. We discuss a simple example of the opposite situation: in this case, a numerical experiment shows a subexponential growth.

BLOW-UP OF THE SOLUTION FOR SEMILINEAR DAMPED WAVE EQUATION WITH TIME-DEPENDENT DAMPING

2012 ◽  
Vol 14 (05) ◽  
pp. 1250034
Author(s):  
JIAYUN LIN ◽  
JIAN ZHAI
Keyword(s):  
Cauchy Problem ◽  
Wave Equation ◽  
Initial Data ◽  
Upper Bound ◽  
Blow Up ◽  
Time Dependent ◽  
Damped Wave Equation ◽  
Large Initial Data ◽  
Power Type ◽  
The Cauchy Problem

We consider the Cauchy problem for the damped wave equation with time-dependent damping and a power-type nonlinearity |u|ρ. For some large initial data, we will show that the solution to the damped wave equation will blow up within a finite time. Moreover, we can show the upper bound of the life-span of the solution.

scholarly journals Solution of the Cauchy problem for a time-dependent Schrödinger equation

2008 ◽  
Vol 49 (7) ◽  
pp. 072102 ◽  
Author(s):  
Maria Meiler ◽  
Ricardo Cordero–Soto ◽  
Sergei K. Suslov
Keyword(s):  
Cauchy Problem ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Time Dependent ◽  
The Cauchy Problem
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