摘要:

Bäcklund transformations, which are relations among solutions of partial differential equations–usually nonlinear–have been found and applied mainly for systems with two independent variables. A few are known for equations like the Kadomtsev-Petviashvili equation [1], which has three independent variables, but they are rare. Wahlquist and Estabrook [2] discovered a systematic method for searching for Bäcklund transformations, using an auxiliary linear system called a prolongation structure. The integrability conditions for the prolongation structure are to be the original differential equation system, most of which systems have just two independent variables. This paper discusses how the Wahlquist-Estabrook method might be applied to systems with larger numbers of variables, with the Kadomtsev-Petviashvili equation as an example. The Zakharov-Shabat method is also discussed. Applications to other equations, such as the Davey-Stewartson and Einstein equation systems, are presented.

ISSN:1402-9251    

DOI:10.2991/jnmp.1995.2.3-4.1

出版商:Taylor & Francis Group    

年代:1995

数据来源: Taylor

摘要:

In this talk I am going to present a brief review of some key ideas and methods which were given start and were developed in Kyiv, at the Institute of Mathematics of National Academy of Sciences of Ukraine during recent years.

ISSN:1402-9251    

DOI:10.2991/jnmp.1995.2.3-4.2

出版商:Taylor & Francis Group    

年代:1995

数据来源: Taylor

摘要:

The Program SUBMODELS [1] is aimed to exhaust all possibilities derived from the symmetry of differential equations for construction of submodels (i.e., systems of equations of the reduced dimension) which describe classes of exact solutions for initial equations. In the frame of this Program, our paper communicates the description of all (up to similarity) regular submodels for the system of Gas Dynamics with the general state equation. Submodels are classified by their types (σ, δ), whereσis a rank andδis a defect. Regularity of a submodel means that invariant independent variables are functions of initial independent variables only. The classification tables, comments to them, and some characteristic examples are presented.

ISSN:1402-9251    

DOI:10.2991/jnmp.1995.2.3-4.3

出版商:Taylor & Francis Group    

年代:1995

数据来源: Taylor

摘要:

We consider a representation of canonical commutation relations (CCR) appearing in a (non-Abelian) gauge theory on a non-simply connected region in the two-dimensional Euclidean space. A necessary and sufficient condition for the representation to be equivalent to the Schrödinger representation of CCR is given in terms of Wilson loops. A representation inequivalent to the Schrödinger representation gives a mathematical expression for the (non-Abelian) Aharonov-Bohm effect. Some aspects of the Dirac-Weyl operator associated with the representation of CCR are discussed.

ISSN:1402-9251    

DOI:10.2991/jnmp.1995.2.3-4.4

出版商:Taylor & Francis Group    

年代:1995

数据来源: Taylor

摘要:

We investigate the structure of certain types of subalgebras of Galilei algebras and the relationship between the conjugacies of these subalgebras under different groups of automorphisms.

ISSN:1402-9251    

DOI:10.2991/jnmp.1995.2.3-4.5

出版商:Taylor & Francis Group    

年代:1995

数据来源: Taylor

摘要:

We study symmetries of the real Maxwell-Bloch equations. We give a Lax pair, bi-Hamiltonian formulations and we find a symplectic realization of the system. We have also constructed a hierarchy of master symmetries which is used to generate nonlinear Poisson brackets. In addition we have calculated the classical Lie point symmetries and variational symmetries.

ISSN:1402-9251    

DOI:10.2991/jnmp.1995.2.3-4.6

出版商:Taylor & Francis Group    

年代:1995

数据来源: Taylor

ISSN:1402-9251    

DOI:10.2991/jnmp.1995.2.3-4.7

出版商:Taylor & Francis Group    

年代:1995

数据来源: Taylor

ISSN:1402-9251    

DOI:10.2991/jnmp.1995.2.3-4.8

出版商:Taylor & Francis Group    

年代:1995

数据来源: Taylor

摘要:

The Madelung representationψ=uexp(iv) is considered for the d’Alembert equation□nψ−F(|ψ|)ψ= 0 to develop a technique for finding exact solutions. We classify the nonlinear functionFfor which the amplitude and phase of the d’Alembert equation are related to the solutions of the compatible d’Alembert–Hamiltonian system.

ISSN:1402-9251    

DOI:10.2991/jnmp.1995.2.3-4.9

出版商:Taylor & Francis Group    

年代:1995

数据来源: Taylor

摘要:

Lie reduction of the Navier-Stokes equations to systems of partial differential equations in three and two independent variables and to ordinary differential equations is described.

ISSN:1402-9251    

DOI:10.2991/jnmp.1995.2.3-4.10

出版商:Taylor & Francis Group    

年代:1995

数据来源: Taylor