Journal of Nonlinear Mathematical Physics

Volume 9, Issue Supplement 1, February 2002

Recent Advances in Integrable Systems

Proceedings of the Special Session on Integrable Systems of the First Joint Meeting of the American Mathematical Society and the Hong Kong Mathematical Society

Research Article

1. From Bi-Hamiltonian Geometry to Separation of Variables: Stationary Harry-Dym and the KdV Dressing Chain

Maciej Blaszak
Pages: 1 - 13
Separability theory of one-Casimir Poisson pencils, written down in arbitrary coordnates, is presented. Separation of variables for stationary Harry-Dym and the KdV dressing chain illustrates the theory.
Research Article

2. A Truncation for Obtaining all the First Degree Birational Transformations of the Painlevé Transcendents

Robert Conte, Micheline Musette
Pages: 14 - 28
A birational transformation is one which leaves invariant an ordinary differential eqution, only changing its parameters. We first recall the consistent truncation which has allowed us to obtain the first degree birational transformation of Okamoto for the mater Painlevé equation P6. Then we improve...
Research Article

3. Bäcklund Transformations on Coadjoint Orbits of the Loop Algebra ~gl(r)

Yuri Fedorov
Pages: 29 - 46
There is a wide class of integrable Hamiltonian systems on finite-dimensional coadjoint orbits of the loop algebra ~gl(r) which are represented by r × r Lax equations with a rational spectral parameter. A reduced complex phase space is foliated with open subsets of Jacobians of regularized spectral curves....
Research Article

4. New Symmetry Reductions for some Ordinary Differential Equations

M.L. Gandarias, E. Medina, C. Muriel
Pages: 47 - 58
In this work we derive potential symmetries for ordinary differential equations. By using these potential symmetries we find that the order of the ODE can be reduced even if this equation does not admit point symmetries. Moreover, in the case for which the ODE admits a group of point symmetries, we find...
Research Article

5. Some Special Integrable Surfaces

M. Gürses
Pages: 59 - 66
We consider surfaces arising from integrable partial differential equations and from their deformations. Symmetries of the equation, gauge transformation of the corrsponding Lax pair and spectral parameter transformations are the deformations which lead infinitely many integrable surfaces. We also study...
Research Article

6. Is My ODE a Painlevé Equation in Disguise?

Jarmo Hietarinta, Valery Dryuma
Pages: 67 - 74
Painlevé equations belong to the class y +a1 y 3 +3a2 y 2 +3a3 y +a4 = 0, where ai = ai(x, y). This class of equations is invariant under the general point transformation x = (X, Y ), y = (X, Y ) and it is therefore very difficult to find out whether two equations in this class are related. We describe...
Research Article

7. Rational Solutions of an Extended Lotka-Volterra Equation

X.B. Hu, P.A. Clarkson
Pages: 75 - 83
A series of rational solutions are presented for an extended Lotka-Volterra eqution. These rational solutions are obtained by using Hirota's bilinear formalism and Bäcklund transformation. The crucial step is the use of nonlinear superposition fomula. The so-called extended Lotka-Volterra equation is...
Research Article

8. Exact Solutions of DNLS and Derivative Reaction-Diffuson Systems

Jyh-Hao Lee, Yen-Ching Lee, Chien-Chih Lin
Pages: 87 - 98
In this paper, we obtain some exact solutions of Derivative Reaction-Diffusion (DRD) system and, as by-products, we also show some exact solutions of DNLS via Hirota bilinearization method. At first, we review some results about two by two AKNS-ZS system, then introduce Hirota bilinearization method...
Research Article

9. Hamiltonian Structure and Linear Stability of Solitary Waves of the Green-Naghdi Equations

Yi A. Li
Pages: 99 - 105
We investigate linear stability of solitary waves of a Hamiltonian system. Unlike weakly nonlinear water wave models, the physical system considered here is nonlinearly dispersive, and contains nonlinearity in its highest derivative term. This results in more detailed asymptotic analysis of the eigenvalue...
Research Article

10. Adjoint Symmetry Constraints Leading to Binary Nonlinearization

Wen-Xiu Ma, Ruguang Zhou
Pages: 106 - 126
Adjoint symmetry constraints are presented to manipulate binary nonlinearization, and shown to be a slight weaker condition than symmetry constraints in the case of Hamiltonian systems. Applications to the multicomponent AKNS system of nonlinear Schrödinger equations and the multi-wave interaction equations,...
Research Article

11. Bilinear Forms of Integrable Lattices Related to Toda and Lotka-Volterra Lattices

Ken-ichi Maruno, Wen-Xiu Ma
Pages: 127 - 139
Hirota's bilinear technique is applied to some integrable lattice systems related to the Bäcklund transformations of the 2DToda, Lotka-Volterra and relativistic LotkVolterra lattice systems, which include the modified Lotka-Volterra lattice system, the modified relativistic Lotka-Volterra lattice system,...
Research Article

12. Intrinsic Characterizations of Orthogonal Separability for Natural Hamiltonians with Scalar Potentials on Pseudo-Riemannian Spaces

Raymond G. McLenaghan, Roman G. Smirnov
Pages: 140 - 151
Orthogonal separability of finite-dimensional Hamiltonians is characterized by using various geometrical concepts, including Killing tensors, moving frames, the Nijehuis tensor, bi-Hamiltonian and quasi-bi-Hamiltonian representations. In addition, a complete classification of separable metrics defined...
Research Article

13. On a q-Analog of ADHMN Construction for Self-Dual Yang-Mills

Atsushi Nakamula
Pages: 152 - 163
It is known that many integrable systems can be reduced from self-dual Yang-Mills equations. The formal solution space to the self-dual Yang-Mills equations is given by the so called ADHM construction, in which the solution space are graded by vector spaces with dimensionality concerning topological...
Research Article

14. Ghost Symmetries

Peter J. Olver, Jan A. Sanders, Jing Ping Wang
Pages: 164 - 172
We introduce the notion of a ghost characteristic for nonlocal differential equations. Ghosts are essential for maintaining the validity of the Jacobi identity for the charateristics of nonlocal vector fields.
Research Article

15. Integrable Systems and Metrics of Constant Curvature

Maxim Pavlov
Pages: 173 - 191
In this article we present a Lagrangian representation for evolutionary systems with a Hamiltonian structure determined by a differential-geometric Poisson bracket of the first order associated with metrics of constant curvature. Kaup-Boussinesq system has three local Hamiltonian structures and one nonlocal...
Research Article

16. A Nonlocal Kac-van Moerbeke Equation Admitting N-Soliton Solutions

Simon Ruijsenaars
Pages: 192 - 206
Using our previous work on reflectionless analytic difference operators and a nonlocal Toda equation, we introduce analytic versions of the Volterra and Kac-van Moerbeke lattice equations. The real-valued N-soliton solutions to our nonlocal equations corrspond to self-adjoint reflectionless analytic...
Research Article

17. A Search for Higher-Dimensional Integrable Modified KdV Equations ­ The Painlevé Approach

Kouichi Toda
Pages: 207 - 212
It is shown here that the possibility of the existence of new (2 + 1) dimensional intgrable equations of the modified KdV equation using the Painlevé test.
Research Article

18. A List of 1 + 1 Dimensional Integrable Equations and Their Properties

Jing Ping Wang
Pages: 213 - 233
This paper contains a list of known integrable systems. It gives their recursion-, Hamiltonian-, symplectic- and cosymplectic operator, roots of their symmetries and their scaling symmetry.
Research Article

19. Linearization of Mirror Systems

Tat Leung Yee
Pages: 234 - 242
We demonstrate, through the fourth Painlevé and the modified KdV equations, that the attempt at linearizing the mirror systems (more precisely, the equation satisfied by the new variable introduced in the indicial normalization) near movable poles can naturally lead to the Schlesinger transformations...
Research Article

20. Deformations of the Bihamiltonian Structures on the Loop Space of Frobenius Manifolds

Youjin Zhang
Pages: 243 - 257
We consider an important class of deformations of the genus zero bihamiltonian struture defined on the loop space of semisimple Frobenius manifolds, and present results on such deformations at the genus one and genus two approximations.