254 research outputs found
An application of the Maslov complex germ method to the 1D nonlocal Fisher-KPP equation
A semiclassical approximation approach based on the Maslov complex germ
method is considered in detail for the 1D nonlocal
Fisher-Kolmogorov-Petrovskii-Piskunov 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-Kolmogorov-Petrovskii-Piskunov 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.Comment: 28 pages, version accepted for publication in Int. J. Geom. Methods
Mod. Phy
The Trajectory-Coherent Approximation and the System of Moments for the Hartree-Type Equation
The general construction of quasi-classically concentrated solutions to the
Hartree-type equation, based on the complex WKB-Maslov method, is presented.
The formal solutions of the Cauchy problem for this equation, asymptotic in
small parameter \h (\h\to0), are constructed with a power accuracy of
O(\h^{N/2}), where N is any natural number. In constructing the
quasi-classically concentrated solutions, a set of Hamilton-Ehrenfest equations
(equations for middle or centered moments) is essentially used. The nonlinear
superposition principle has been formulated for the class of quasi-classically
concentrated solutions of the Hartree-type equations. The results obtained are
exemplified by the one-dimensional equation Hartree-type with a Gaussian
potential.Comments: 6 pages, 4 figures, LaTeX Report no: Subj-class:
Accelerator PhysicsComment: 36 pages, LaTeX-2
Symmetry Operators for the Fokker-Plank-Kolmogorov Equation with Nonlocal Quadratic Nonlinearity
The Cauchy problem for the Fokker-Plank-Kolmogorov equation with a nonlocal
nonlinear drift term is reduced to a similar problem for the correspondent
linear equation. The relation between symmetry operators of the linear and
nonlinear Fokker-Plank-Kolmogorov equations is considered. Illustrative
examples of the one-dimensional symmetry operators are presented.Comment: This is a contribution to the Vadim Kuznetsov Memorial Issue on
Integrable Systems and Related Topics, published in SIGMA (Symmetry,
Integrability and Geometry: Methods and Applications) at
http://www.emis.de/journals/SIGMA
Symmetry and Intertwining Operators for the Nonlocal Gross-Pitaevskii Equation
We consider the symmetry properties of an integro-differential
multidimensional Gross-Pitaevskii equation with a nonlocal nonlinear (cubic)
term in the context of symmetry analysis using the formalism of semiclassical
asymptotics. This yields a semiclassically reduced nonlocal Gross-Pitaevskii
equation, which can be treated as a nearly linear equation, to determine the
principal term of the semiclassical asymptotic solution. Our main result is an
approach which allows one to construct a class of symmetry operators for the
reduced Gross-Pitaevskii equation. These symmetry operators are determined by
linear relations including intertwining operators and additional algebraic
conditions. The basic ideas are illustrated with a 1D reduced Gross-Pitaevskii
equation. The symmetry operators are found explicitly, and the corresponding
families of exact solutions are obtained
Photon recoil momentum in a Bose-Einstein condensate of a dilute gas
We develop a "minimal" microscopic model to describe a
two-pulse-Ramsay-interferometer-based scheme of measurement of the photon
recoil momentum in a Bose-Einstein condensate of a dilute gas [Campbell et al.,
Phys. Rev. Lett. 94, 170403 (2005)]. We exploit the truncated coupled
Maxwell-Schroedinger equations to elaborate the problem. Our approach provides
a theoretical tool to reproduce essential features of the experimental results.
Additionally, we enable to calculate the quantum-mechanical mean value of the
recoil momentum and its statistical distribution that provides a detailed
information about the recoil event.Comment: 6 pages, 4 figure
- …