1,508 research outputs found
Classification of classical and non-local symmetries of fourth-order nonlinear evolution equations
In this paper, we consider group classification of local and quasi-local
symmetries for a general fourth-order evolution equations in one spatial
variable. Following the approach developed by Zhdanov and Lahno, we construct
all inequivalent evolution equations belonging to the class under study which
admit either semi-simple Lie groups or solvable Lie groups. The obtained lists
of invariant equations (up to a local change of variables) contain both the
well-known equations and a variety of new ones possessing rich symmetry. Based
on the results on the group classification for local symmetries, the group
classification for quasi-local symmetries of the equations is also given.Comment: LaTeX, 60 page
The Structure of Lie Algebras and the Classification Problem for Partial Differential Equations
The present paper solves completely the problem of the group classification
of nonlinear heat-conductivity equations of the form\
. We have proved, in particular,
that the above class contains no nonlinear equations whose invariance algebra
has dimension more than five. Furthermore, we have proved that there are two,
thirty-four, thirty-five, and six inequivalent equations admitting one-, two-,
three-, four- and five-dimensional Lie algebras, respectively. Since the
procedure which we use, relies heavily upon the theory of abstract Lie algebras
of low dimension, we give a detailed account of the necessary facts. This
material is dispersed in the literature and is not fully available in English.
After this algebraic part we give a detailed description of the method and then
we derive the forms of inequivalent invariant evolution equations, and compute
the corresponding maximal symmetry algebras. The list of invariant equations
obtained in this way contains (up to a local change of variables) all the
previously-known invariant evolution equations belonging to the class of
partial differential equations under study.Comment: 45 page
Theory and Design of Flight-Vehicle Engines
Papers are presented on such topics as the testing of aircraft engines, errors in the experimental determination of the parameters of scramjet engines, the effect of the nonuniformity of supersonic flow with shocks on friction and heat transfer in the channel of a hypersonic ramjet engine, and the selection of the basic parameters of cooled GTE turbines. Consideration is also given to the choice of optimal total wedge angle for the acceleration of aerospace vehicles, the theory of an electromagnetic-resonator engine, the dynamic characteristics of the pumps and turbines of liquid propellant rocket engines in transition regimes, and a hierarchy of mathematical models for spacecraft control engines
The stationary KdV hierarchy and so(2,1) as a spectrum generating algebra
The family F of all potentials V(x) for which the Hamiltonian H in one space
dimension possesses a high order Lie symmetry is determined. A sub-family F',
which contains a class of potentials allowing a realization of so(2,1) as
spectrum generating algebra of H through differential operators of finite
order, is identified. Furthermore and surprisingly, the families F and F' are
shown to be related to the stationary KdV hierarchy. Hence, the "harmless"
Hamiltonian H connects different mathematical objects, high order Lie symmetry,
realization of so(2,1)-spectrum generating algebra and families of nonlinear
differential equations. We describe in a physical context the interplay between
these objects.Comment: 15 pages, LaTe
Conditional symmetry and spectrum of the one-dimensional Schr\"odinger equation
We develop an algebraic approach to studying the spectral properties of the
stationary Schr\"odinger equation in one dimension based on its high order
conditional symmetries. This approach makes it possible to obtain in explicit
form representations of the Schr\"odinger operator by matrices for
any and, thus, to reduce a spectral problem to a purely
algebraic one of finding eigenvalues of constant matrices. The
connection to so called quasi exactly solvable models is discussed. It is
established, in particular, that the case, when conditional symmetries reduce
to high order Lie symmetries, corresponds to exactly solvable Schr\"odinger
equations. A symmetry classification of Sch\"odinger equation admitting
non-trivial high order Lie symmetries is carried out, which yields a hierarchy
of exactly solvable Schr\"odinger equations. Exact solutions of these are
constructed in explicit form. Possible applications of the technique developed
to multi-dimensional linear and one-dimensional nonlinear Schr\"odinger
equations is briefly discussed.Comment: LaTeX-file, 31 pages, to appear in J.Math.Phys., v.37, N7, 199
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
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