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
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.
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...
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...
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...
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...
Pages: 67 - 74
Painlevé equations belong to the class y +a1 y
+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...
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...
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...
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...
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...
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,...
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...
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...
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.
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...
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...
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.
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.
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...
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.