The complex germ method in fock space. I. Wave packet type asymptotics

1995 ◽  
Vol 104 (2) ◽  
pp. 1013-1028 ◽  
Author(s):  
V. P. Maslov ◽  
O. Yu. Shvedov
Keyword(s):  
Wave Packet ◽  
Fock Space ◽  
Complex Germ ◽  
Complex Germ Method

Maslov's complex germ method and Berry's phase

1994 ◽  
Vol 27 (18) ◽  
pp. 6267-6286 ◽  
Author(s):  
A Yu Trifonov ◽  
A A Yevseyevich
Keyword(s):  
Complex Germ ◽  
Berry's Phase ◽  
Berry’S Phase ◽  
Complex Germ Method

Quantization of closed orbits in Dirac theory by Maslov's complex germ method

1994 ◽  
Vol 27 (3) ◽  
pp. 1021-1043 ◽  
Author(s):  
V G Bagrov ◽  
V V Belov ◽  
A Yu Trifonov ◽  
A A Yevseyevich
Keyword(s):  
Dirac Theory ◽  
Complex Germ ◽  
Closed Orbits ◽  
Complex Germ Method

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.

scholarly journals A complete set of Lorentz-invariant wave packets and modified uncertainty relation

2021 ◽  
Vol 81 (8) ◽  
Author(s):  
Kin-ya Oda ◽  
Juntaro Wada
Keyword(s):  
Wave Packet ◽  
Uncertainty Relation ◽  
Fock Space ◽  
Wave Packets ◽  
Completeness Relation ◽  
Relativistic Limit ◽  
Heisenberg Uncertainty Principle ◽  
Heisenberg Uncertainty ◽  
Complete Set ◽  
Hilbert Subspace

AbstractWe define a set of fully Lorentz-invariant wave packets and show that it spans the corresponding one-particle Hilbert subspace, and hence the whole Fock space as well, with a manifestly Lorentz-invariant completeness relation (resolution of identity). The position–momentum uncertainty relation for this Lorentz-invariant wave packet deviates from the ordinary Heisenberg uncertainty principle, and reduces to it in the non-relativistic limit.

scholarly journals Short-Wave Asymptotics for Gaussian Beams and Packets and Scalarization of Equations in Plasma Physics

Physics ◽  
2019 ◽  
Vol 1 (2) ◽  
pp. 301-320
Author(s):  
Anatoly Yu. Anikin ◽  
Sergey Yu. Dobrokhotov ◽  
Alexander I. Klevin ◽  
Brunello Tirozzi
Keyword(s):  
Wave Packet ◽  
High Frequency ◽  
Classical Mechanics ◽  
Short Wave ◽  
Lower Hybrid ◽  
Focal Points ◽  
Complex Germ ◽  
High Frequency Waves ◽  
Arbitrary Frequency ◽  
Main Ideas

We study Gaussian wave beam and wave packet types of solutions to the linearized cold plasma system in a toroidal domain (tokamak). Such solutions are constructed with help of Maslov’s complex germ theory (short-wave or semi-classical asymptotics with complex phases). The term “semi-classical” asymptotics is understood in a broad sense: asymptotic solutions of evolutionary and stationary partial differential equations from wave or quantum mechanics are expressed through solutions of the corresponding equations of classical mechanics. This, in particular, allows one to use useful geometric considerations. The small parameter of the expansion is h = λ / 2 π L where λ is the wavelength and L the dimension of the system. In order to apply the asymptotic algorithm, we need this parameter to be small, so we deal only with high-frequency waves, which are in the range of lower hybrid waves used to heat the plasma. The asymptotic solution appears to be a Gaussian wave packet divided by the square root of the determinant of an appropriate Jacobi matrix (“complex divergence”). When this determinant is zero, focal points appear. Our approach allows one to write out asymptotics near focal points. We also claim that this approach is very practical and leads to formulas that can be used for numerical simulations in software like Wolfram Mathematica, Maple, etc. For the particular case of high-frequency beams, we present a recipe for constructing beams and packets and show the results of their numerical implementation. We also propose ideas to treat the more difficult general case of arbitrary frequency. We also explain the main ideas of asymptotic theory used to obtain such formulas.

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