Journal of Nonlinear Mathematical Physics

Volume 8, Issue 1, February 2001
Research Article

1. UV Manifold and Integrable Systems in Spaces of Arbitrary Dimension

A.N. Leznov
Pages: 1 - 7
The 2n dimensional manifold with two mutually commutative operators of differetiation is introduced. Nontrivial multidimensional integrable systems connected with arbitrary graded (semisimple) algebras are constructed. The general solution of them is presented in explicit form.
Research Article

2. Correctors for Some Nonlinear Monotone Operators

Johan Byström
Pages: 8 - 30
In this paper we study homogenization of quasi-linear partial differential equations of the form -div (a (x, x/h, Duh)) = fh on with Dirichlet boundary conditions. Here the sequence (h) tends to 0 as h and the map a (x, y, ) is periodic in y, monotone in and satisfies suitable continuity conditions....
Research Article

3. On Bilinear Invariant Differential Operators Acting on Tensor Fields on the Symplectic Manifold

Pavel Grozman
Pages: 31 - 37
Let M be an n-dimensional manifold, V the space of a representation : GL(n)GL(V ). Locally, let T(V ) be the space of sections of the tensor bundle with fiber V over a sufficiently small open set U M, in other words, T(V ) is the space of tensor fields of type V on M on which the group Diff(M) of diffeomorphisms...
Research Article

4. Fractal and Chaotic Solutions of the Discrete Nonlinear Schrödinger Equation in Classical and Quantum Systems

H.S. Dhillon, F.V. Kusmartsev, K.E. Kürten
Pages: 38 - 49
We discuss stationary solutions of the discrete nonlinear Schrödinger equation (DNSE) with a potential of the 4 type which is generically applicable to several quantum spin, electron and classical lattice systems. We show that there may arise chaotic spatial structures in the form of incommensurate or...
Short Communication

5. On Algebraic Integrability of the Deformed Elliptic Calogero­Moser Problem

L.A. Khodarinova, I.A. Prikhodski
Pages: 50 - 53
Algebraic integrability of the elliptic Calogero­Moser quantum problem related to the deformed root systems A2(2) is proved. Explicit formulae for integrals are found.
Research Article

6. Orthogonalization of Graded Sets of Vectors

I.A. Shreshevskii
Pages: 54 - 58
I propose an orthogonalization procedure preserving the grading of the initial graded set of linearly independent vectors. In particular, this procedure is applicable for orthonormalization of any countable set of polynomials in several (finitely many) ideterminates.
Research Article

7. Superanalogs of the Calogero Operators and Jack Polynomials

A. Sergeev
Pages: 59 - 64
A depending on a complex parameter k superanalog SL of Calogero operator is costructed; it is related with the root system of the Lie superalgebra gl(n|m). For m = 0 we obtain the usual Calogero operator; for m = 1 we obtain, up to a change of indterminates and parameter k the operator constructed by...
Research Article

8. Hard Loss of Stability in Painlevé-2 Equation

O.M. Kiselev
Pages: 65 - 95
A special asymptotic solution of the Painlevé-2 equation with small parameter is stdied. This solution has a critical point t corresponding to a bifurcation phenomenon. When t < t the constructed solution varies slowly and when t > t the solution oscillates very fast. We investigate the transitional...
Research Article

9. Integrability and Explicit Solutions in Some Bianchi Cosmological Dynamical Systems

J. Chavarriga, I.A. Garcia
Pages: 96 - 105
The Einstein field equations for several cosmological models reduce to polynomial systems of ordinary differential equations. In this paper we shall concentrate our attention to the spatially homogeneous diagonal G2 cosmologies. By using Darboux's theory in order to study ordinary differential equations...
Research Article

10. Reflectionless Analytic Difference Operators I. Algebraic Framework

S.N.M. Ruijsenaars
Pages: 106 - 138
We introduce and study a class of analytic difference operators admitting reflectionless eigenfunctions. Our construction of the class is patterned after the Inverse Scattering Transform for the reflectionless self-adjoint Schrödinger and Jacobi operators corrsponding to KdV and Toda lattice solitons.
Research Article

11. Symmetry, Singularities and Integrability in Complex Dynamics III: Approximate Symmetries and Invariants

P.G.L. Leach, S. Moyo, S. Cotsakis, R.L. Lemmer
Pages: 139 - 156
The different natures of approximate symmetries and their corresponding first intgrals/invariants are delineated in the contexts of both Lie symmetries of ordinary differential equations and Noether symmetries of the Action Integral. Particular note is taken of the effect of taking higher orders of the...
Review Article

12. The Maupertuis Principle and Canonical Transformations of the Extended Phase Space

A.V. Tsiganov
Pages: 157 - 182
We discuss some special classes of canonical transformations of the extended phase space, which relate integrable systems with a common Lagrangian submanifold. Vrious parametric forms of trajectories are associated with different integrals of motion, Lax equations, separated variables and action-angles...