Journal of Nonlinear Mathematical Physics

Volume 7, Issue 1, February 2000

1. Lax Pairs, Painlevé Properties and Exact Solutions of the Calogero Korteweg-de Vries Equation and a New (2 + 1)-Dimensional Equation

Song-Ju YU, Kouichi TODA
Pages: 1 - 13
We prove the existence of a Lax pair for the Calogero Korteweg-de Vries (CKdV) equation. Moreover, we modify the T operator 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...

2. Links Between Different Analytic Descriptions of Constant Mean Curvature Surfaces

Pages: 14 - 21
Transformations between different analytic descriptions of constant mean curvature (CMC) surfaces are established. In particular, it is demonstrated that the system

3. On the Calculation of Finite-Gap Solutions of the KdV Equation

Pages: 22 - 33
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...

4. On the Structure of the Bäcklund Transformations for the Relativistic Lattices

Vsevolod E. ADLER
Pages: 34 - 56
The Bäcklund 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 Bäcklund...

5. On Exact Solution of a Classical 3D Integrable Model

Pages: 57 - 72
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...

6. q-Probability: I. Basic Discrete Distributions

Pages: 73 - 93
For basic discrete probability distributions, - Bernoulli, Pascal, Poisson, hypergemetric, contagious, and uniform, - q-analogs are proposed.

7. Particles and Strings in a 2 + 1-D Integrable Quantum Model

Pages: 94 - 119
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 Yang­Baxter equation. We construct...