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 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 An application of the Maslov complex germ method to the one-dimensional nonlocal Fisher–KPP equation

2018 ◽  
Vol 15 (06) ◽  
pp. 1850102 ◽  
Author(s):  
A. V. Shapovalov ◽  
A. Yu. Trifonov
Keyword(s):  
Nonlinear Equation ◽  
Linear Equation ◽  
Algebraic Equations ◽  
Semiclassical Asymptotics ◽  
Complex Germ ◽  
Kpp Equation ◽  
One Dimensional ◽  
Maslov Complex Germ ◽  
The One ◽  
Complex Germ Method

A semiclassical approximation approach based on the Maslov complex germ method is considered in detail for the one-dimensional nonlocal Fisher–Kolmogorov–Petrovskii–Piskunov (Fisher–KPP) equation under the supposition of weak diffusion. In terms of the semiclassical formalism developed, the original nonlinear equation is reduced to an associated linear partial differential equation and some algebraic equations for the coefficients of the linear equation with a given accuracy of the asymptotic parameter. The solutions of the nonlinear equation are constructed from the solutions of both the linear equation and the algebraic equations. The solutions of the linear problem are found with the use of symmetry operators. A countable family of the leading terms of the semiclassical asymptotics is constructed in explicit form. The semiclassical asymptotics are valid by construction in a finite time interval. We construct asymptotics which are different from the semiclassical ones and can describe evolution of the solutions of the Fisher–KPP equation at large times. In the example considered, an initial unimodal distribution becomes multimodal, which can be treated as an example of a space structure.

SIMPLE AND COMPLEX DYNAMICS FOR ONE-DIMENSIONAL MANIFOLD MAPS

1995 ◽  
Vol 05 (05) ◽  
pp. 1351-1355
Author(s):  
VLADIMIR FEDORENKO
Keyword(s):  
Topological Entropy ◽  
Complex Dynamics ◽  
Periodic Points ◽  
Dimensional Manifold ◽  
One Dimensional ◽  
Circle Maps ◽  
Interval Maps ◽  
Chain Recurrent Set ◽  
Recurrent Set

We give a characterization of complex and simple interval maps and circle maps (in the sense of positive or zero topological entropy respectively), formulated in terms of the description of the dynamics of the map on its chain recurrent set. We also describe the behavior of complex maps on their periodic points.

scholarly journals Polymer quantum effects on compact stars models

2015 ◽  
Vol 24 (05) ◽  
pp. 1550033 ◽  
Author(s):  
Guillermo Chacón-Acosta ◽  
Héctor H. Hernandez-Hernandez
Keyword(s):  
Semiclassical Approximation ◽  
Generalized Uncertainty Principle ◽  
Compact Object ◽  
Compact Stars ◽  
One Dimensional ◽  
Degenerate Fermi Gas ◽  
Mechanical Models ◽  
Degeneracy Pressure ◽  
Loop Quantization ◽  
Bose Einstein

In this work we study a completely degenerate Fermi gas at zero temperature by a semiclassical approximation for a Hamiltonian that arises in polymer quantum mechanics. Polymer quantum systems are quantum mechanical models quantized in a similar way as in loop quantum gravity, allowing the study of the discreteness of space and other features of the loop quantization in a simplified way. We obtain the polymer modified thermodynamical properties for this system by noticing that the corresponding Fermi energy is exactly the same as if one directly polymerizes the momentum pF. We also obtain the expansion of the corresponding thermodynamical variables in terms of small values of the polymer length scale λ. We apply these results to study a simple model of a compact one-dimensional star where the gravitational collapse is supported by electron degeneracy pressure. As a consequence, polymer corrections to the mass of the object are found. By using bounds for the polymer length found in Bose–Einstein condensates experiments we compute the modification in the mass of the compact object due to polymer effects of order ~ 10-8. This result is similar to the other order found by different approaches such as generalized uncertainty principle (GUP), and that certainly is within the error reported in typical measurements of white dwarf masses.

Computation of 2D Manifold Based on Generalized Condition

2012 ◽  
Vol 580 ◽  
pp. 210-213
Author(s):  
Zhong Wu ◽  
Meng Jia ◽  
Qing Hua Ji
Keyword(s):  
Dimensional Manifold ◽  
One Dimensional ◽  
Prediction Scheme ◽  
Accuracy Criterion

The new algorithm uses the idea of growing the manifold. The preimage of the new point is found quickly with a method called gradient prediction scheme and a new accuracy criterion is proposed. Furthermore, our algorithm is capable of computing both stable and unstable one dimensional manifold.

On the stability of lattice boltzmann equations for one-dimensional diffusion equation

2017 ◽  
Vol 08 (01) ◽  
pp. 1750013 ◽  
Author(s):  
Gerasim Vladimirovich Krivovichev
Keyword(s):  
Lattice Boltzmann ◽  
Initial Conditions ◽  
Kinetic Equations ◽  
Differential Approximation ◽  
Boltzmann Equations ◽  
Von Neumann ◽  
One Dimensional ◽  
Dimensional Diffusion ◽  
The Cauchy Problem ◽  
Lattice Boltzmann Equations

Stability analysis of lattice Boltzmann equations (LBEs) on initial conditions for one-dimensional diffusion is performed. Stability of the solution of the Cauchy problem for the system of linear Bhatnaghar–Gross–Krook kinetic equations is demonstrated for the cases of D1Q2 and D1Q3 lattices. Stability of the scheme for D1Q2 lattice is analytically analyzed by the method of differential approximation. Stability of parametrical scheme is numerically investigated by von Neumann method in parameter space. As a result of numerical analysis, the correction of the hypothesis on transfer of stability conditions of the scheme for macroequation to the system of LBEs is demonstrated.

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