12 research outputs found
Group classification of variable coefficient quasilinear reaction-diffusion equations
The group classification of variable coefficient quasilinear
reaction-diffusion equations is carried out exhaustively.
This became possible due to usage of a conditional equivalence group found in
the course of the study of admissible point transformation within the class.Comment: 10 pages, submitted to the Proceedings of the XVII Geometrical
Seminar (September 3-8, 2012, Zlatibor, Serbia
Quantum integrable systems in three-dimensional magnetic fields: the Cartesian case
In this paper we construct integrable three-dimensional quantum-mechanical
systems with magnetic fields, admitting pairs of commuting second-order
integrals of motion. The case of Cartesian coordinates is considered. Most of
the systems obtained are new and not related to the separation of variables in
the corresponding Schr\"odinger equation.Comment: 8 page
On separable Schr\"odinger equations
We classify (1+3)-dimensional Schr\"odinger equations for a particle
interacting with the electromagnetic field that are solvable by the method of
separation of variables. As a result, we get eleven classes of the
electromagnetic vector potentials of the electromagnetic field , providing separability of the
corresponding Schr\"odinger equations. It is established, in particular, that
the necessary condition for the Schr\"odinger equation to be separable is that
the magnetic field must be independent of the spatial variables. Next, we prove
that any Schr\"odinger equation admitting variable separation into second-order
ordinary differential equations can be reduced to one of the eleven separable
Schr\"odinger equations mentioned above and carry out variable separation in
the latter. Furthermore, we apply the results obtained for separating variables
in the Hamilton-Jacobi equation.Comment: 30 pages, LaTe
On separable Fokker-Planck equations with a constant diagonal diffusion matrix
We classify (1+3)-dimensional Fokker-Planck equations with a constant
diagonal diffusion matrix that are solvable by the method of separation of
variables. As a result, we get possible forms of the drift coefficients
providing separability of the
corresponding Fokker-Planck equations and carry out variable separation in the
latter. It is established, in particular, that the necessary condition for the
Fokker-Planck equation to be separable is that the drift coefficients must be linear. We also find the necessary condition for
R-separability of the Fokker-Planck equation. Furthermore, exact solutions of
the Fokker-Planck equation with separated variables are constructedComment: 20 pages, LaTe
On separable Schrödinger equations
We classify (1+3)-dimensional Schrodinger equations for a particle interacting with the electro-magnetic field that are solvable by the method of separation of variables. As a result, we get eleven classes of the vector-potentials of the electro-magnetic field A(t; ~x) = (A 0 (t; ~x), ~ A(t; ~x)) providing separability of the corresponding Schrodinger equations