arXiv subjects
Publications and source records attributed to A. N. Leznov.
In the present paper it is considered space time manifold with four commutative coordinates with Poincáre group of motion but different from Poincáre Minkowski space time of the modern physics.
Solution of Monge equation of arbitrary degree (non linear differential equation n-orden) is connected with solution of functional equation for 4 functions with 4 different arguments. Some number solutions of this equation is represented in explicit form.
A way to obtain the series solutions of the 1 + 2 dimensional continuous Toda chain is presented.
It is shown that corresponding to Plebansky equation of symmetry posses the infinite set of solutions, which we present in explicit form. This fact leads to conclusion about possibility to find series solutions of the Plebansky equation in analytic form. Some classes of explicit solution are presented.
The matrix elements connected non physic angles space representation with the space of physic observables of alternative O(2,4) world found on explicit form. All such matrix elements are represented in terms of solutions of Gauss hypergeometric equations. From these results wave function of free moving particle in alternative world may be constructed. Definite relations between micro and macro label variables are presented.
It is proposed four dimensional curved space-time with de-Sitter group of motion. Theory contain free dimension constants of length, impulse and action. Under infinite values of these parameters theory pass to usual Minkowski space-time with Poincare group of motion. In modified space gauge invariant field theory is constructed.
It is shown that algebra of quantum space of the title of the present paper may be realized on usual unphysical Minkowskii one. Equations of field theory and there solutions are discussed. Solution equations of particle motion are obtained in implicit form from the classical ones.
The solution of symmetry equation of Yang-Mills self dual system is found in explicit form of its raising Hamiltonian operator. Thus explicit form of equations of self dual Yang Mills hierarchy is constructed.
Two Poisson structures invariant with respect to discrete transformation of the Maximal root in the case of arbitrary semi-simple algebras are presented in explicit form. Thus the problem of construction of equations of n-wave hierarchy in the case of arbitrary semi simple algebra is solved finally.
It is shown that Kepler problem in deformed (quantum) four-dimensional space in non relativistic limit is integrable in quadratures. In non relativistic limit group of motion of quantum space coincide with Galilei one.
It is shown that trajectories of free motion of the particles in deformed ("quantum") four dimensional space-time are quadratic curves.
By the method of discrete transformation equations of 3-th wave hierarchy are constructed. We present in explicit form two Poisson structures, which allow to construct Hamiltonian operator consequent application of which leads to all equations of this hierarchy. For calculations it will be necessary results of previous paper \cite{1}, which for convenience of the reader we present in corresponding place of the text. The obtained formulae are checked by independent calculations.
With each semisimple algebra it is possible to connect the system of interacting waves. The number of interacting fields coincides with the number of positive roots of corresponding semisimple algebra. Multisoliton solution of such kind problem are represented in explicit form for the algebras of second rank.
By the method of discrete transformation equations of 3-th wave hierarchy are constructed. The main difference compare with the systems connected with $A_1$ algebra consists in a fact that in $A_2$ case there are two different systems of equations of the same degree (the maximal derivatives with respect to space coordinate). In the present paper we construct only system of equations of zero, first and second degree. At this moment the general method for solution of this problem (the type of canonical Hamiltonian operators) is unknown for the author and our direct calculationsare sufficiently combersome.
Multi-soliton solution of the 3-waves problem is represented in explicit determinative form.
Symmetry of SDYM equations for "physically restricted solution" with hermitian group element $G=G^H$ in representation of Yang is described. Such transformation $D^B$ pass some PRS to the new one of the same kind. Transformation contain $2N_G$ arbirtrary functions of two independent arguments, where $N_G$ dimension of the gauge semi-simple algebra. These functions may be considered as additional parameters typical for Backlund aproach and by this reason we use term Backlund for this transformation.
It is shown that there exists two inner authomorpism which lead to different form of the sistems equations of integrable hierarchy. We present discrete and Backlund transformation connected with such systems and a general formula for multi-soliton solutions based on these symmetries.
It is shown uniquely that quantized spaces are realised on four-dimensional compact manifolds. In the case of O(1,5) quantized space this are four independent parameters of O(5) unit vector; in the case of O(2,4) these are parameters of one two-dimensional unit vector (1 parameter) and components of unit four-dimensional vector (3- parameters) and at last in the case O(3,3) these are parameters of 2 independent 3-dimensional unit vectors (each have 2 parameters). This result follows directly only from the condition to have a correct limit to usual theory (correspondence principle).