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\ $u_{t}=F(t,x,u,u_{x})u_{xx} + G(t,x,u,u_{x})$. 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 $n\times n$ matrices for any $n\in {\bf N}$ and, thus, to reduce a spectral problem to a purely algebraic one of finding eigenvalues of constant $n\times n$ 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 $B_1(\vec x),B_2(\vec x),B_3(\vec x)$ 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 $\vec B(\vec x)$ 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