摘要:

We prove the existence of a Lax pair for the Calogero Korteweg-de Vries (CKdV) equation. Moreover, we modify theToperator in the the Lax pair of the CKdV equation, in the search of a (2+1)-dimensional case and thereby propose a new equation in (2+1) dimensions. We named this the (2+1)-dimensional CKdV equation. We show that the CKdV equation as well as the (2+1)-dimensional CKdV equation are integrable in the sense that they possess the Painlevé property. Some exact solutions are also constructed.

ISSN:1402-9251    

DOI:10.2991/jnmp.2000.7.1.1

出版商:Taylor & Francis Group    

年代:2000

数据来源: Taylor

摘要:

Transformations between different analytic descriptions of constant mean curvature (CMC) surfaces are established. In particular, it is demonstrated that the systemdescriptive of CMC surfaces within the framework of the generalized Weierstrass representation, decouples into a direct sum of the elliptic Sh-Gordon and Laplace equations. Connections of this system with the sigma model equations are established. It is pointed out, that the instanton solutions correspond to different Weierstrass parametrizations of the standard sphereS2⊂E3.

ISSN:1402-9251    

DOI:10.2991/jnmp.2000.7.1.2

出版商:Taylor & Francis Group    

年代:2000

数据来源: Taylor

摘要:

A simple and general approach for calculating the elliptic finite-gap solutions of the Korteweg-de Vries (KdV) equation is proposed. Our approach is based on the use of the finite-gap equations and the general representation of these solutions in the form of rational functions of the elliptic Weierstrass function. The calculation of initial elliptic finite-gap solutions is reduced to the solution of the finite-band equations with respect to the parameters of the representation. The time evolution of these solutions is described via the dynamic equations of their poles, integrated with the help of the finite-gap equations. The proposed approach is applied by calculating the elliptic 1-, 2- and 3-gap solutions of the KdV equations.

ISSN:1402-9251    

DOI:10.2991/jnmp.2000.7.1.3

出版商:Taylor & Francis Group    

年代:2000

数据来源: Taylor

摘要:

The B¨acklund transformations for the relativistic lattices of the Toda type and their discrete analogues can be obtained as the composition of two duality transformations. The condition of invariance under this composition allows to distinguish effectively the integrable cases. Iterations of the B¨acklund transformations can be described in the terms of nonrelativistic lattices of the Toda type. Several multifield generalizations are presented.

ISSN:1402-9251    

DOI:10.2991/jnmp.2000.7.1.4

出版商:Taylor & Francis Group    

年代:2000

数据来源: Taylor

摘要:

We investigate some classical evolution model in the discrete 2+1 space-time. A map, giving an one-step time evolution, may be derived as the compatibility condition for some systems of linear equations for a set of auxiliary linear variables. Dynamical variables for the evolution model are the coefficients of these systems of linear equations. Determinant of any system of linear equations is a polynomial of two numerical quasimomenta of the auxiliary linear variables. For one, this determinant is the generating functions of all integrals of motion for the evolution, and on the other hand it defines a high genus algebraic curve. The dependence of the dynamical variables on the space-time point (exact solution) may be expressed in terms of theta functions on the jacobian of this curve. This is the main result of our paper.

ISSN:1402-9251    

DOI:10.2991/jnmp.2000.7.1.5

出版商:Taylor & Francis Group    

年代:2000

数据来源: Taylor

摘要:

For basic discrete probability distributions,−Bernoulli, Pascal, Poisson, hypergeometric, contagious, and uniform,−q-analogs are proposed.

ISSN:1402-9251    

DOI:10.2991/jnmp.2000.7.1.6

出版商:Taylor & Francis Group    

年代:2000

数据来源: Taylor

摘要:

We give a review of some recent work on generalization of the Bethe ansatz in the case of 2 + 1-dimensional models of quantum field theory. As such a model, we consider one associated with the tetrahedron equation, i.e. the 2+1-dimensional generalization of the famous Yang–Baxter equation. We construct some eigenstates of the transfer matrix of that model. There arise, together with states composed of point-like particles analogous to those in the usual 1 + 1-dimensional Bethe ansatz, new string-like states and string-particle hybrids.

ISSN:1402-9251    

DOI:10.2991/jnmp.2000.7.1.7

出版商:Taylor & Francis Group    

年代:2000

数据来源: Taylor