8 research outputs found
Iso-spectral deformations of general matrix and their reductions on Lie algebras
We study an iso-spectral deformation of general matrix which is a natural
generalization of the Toda lattice equation. We prove the integrability of the
deformation, and give an explicit formula for the solution to the initial value
problem. The formula is obtained by generalizing the orthogonalization
procedure of Szeg\"{o}. Based on the root spaces for simple Lie algebras, we
consider several reductions of the hierarchy. These include not only the
integrable systems studied by Bogoyavlensky and Kostant, but also their
generalizations which were not known to be integrable before. The behaviors of
the solutions are also studied. Generically, there are two types of solutions,
having either sorting property or blowing up to infinity in finite time.Comment: 25 pages, AMSLaTe
The Poisson bracket compatible with the classical reflection equation algebra
We introduce a family of compatible Poisson brackets on the space of polynomial matrices, which contains the reflection equation algebra bracket.
Then we use it to derive a multi-Hamiltonian structure for a set of integrable
systems that includes the Heisenberg magnet with boundary conditions, the
generalized Toda lattices and the Kowalevski top.Comment: 13 pages, LaTeX with AmsFont
The last integrable case of kozlov-Treshchev Birkhoff integrable potentials
We establish the integrability of the last open case in the Kozlov-Treshchev
classification of Birkhoff integrable Hamiltonian systems. The technique used
is a modification of the so called quadratic Lax pair for Toda lattice
combined with a method used by M. Ranada in proving the integrability of the
Sklyanin case.Comment: 13 page
On Effective Superpotentials and Compactification to Three Dimensions
We study four dimensional N=2 SO/SP supersymmetric gauge theory on R^3\times
S^1 deformed by a tree level superpotential. We will show that the exact
superpotential can be obtained by making use of the Lax matrix of the
corresponding integrable model which is the periodic Toda lattice. The
connection between vacua of SO(2N) and SO(2kN-2k+2) can also be seen in this
framework. Similar analysis can also be applied for SO(2N+1) and SP(2N).Comment: 18 pages, latex file, v2: typos corrected, refs adde
Systems of Hess-Appel'rot type
We construct higher-dimensional generalizations of the classical
Hess-Appel'rot rigid body system. We give a Lax pair with a spectral parameter
leading to an algebro-geometric integration of this new class of systems, which
is closely related to the integration of the Lagrange bitop performed by us
recently and uses Mumford relation for theta divisors of double unramified
coverings. Based on the basic properties satisfied by such a class of systems
related to bi-Poisson structure, quasi-homogeneity, and conditions on the
Kowalevski exponents, we suggest an axiomatic approach leading to what we call
the "class of systems of Hess-Appel'rot type".Comment: 40 pages. Comm. Math. Phys. (to appear
Second order integrability conditions for difference equations. An integrable equation.
Integrability conditions for difference equations admitting a second order formal recursion operator are presented and the derivation of symmetries and canonical conservation laws is discussed. In the generic case, nonlocal conservation laws are also generated. A new integrable equation satis- fying the second order integrability conditions is presented and its integrability is established by the construction of symmetries, conservation laws and a 3Ă—3 Lax representation. Finally, the relation of the symmetries of this equation to a generalized Bogoyavlensky lattice and a new integrable lattice are derived
Integrable Systems
Integrable systems which do not have an \u201cobvious\u201c group symmetry, beginning with the results of Poincar\ue9 and Bruns at the end of the last century, have been perceived as something exotic. The very insignificant list of such examples practically did not change until the 1960\u2019s. Although a number of fundamental methods of mathematical physics were based essentially on the perturbation-theory analysis of the simplest integrable examples, ideas about the structure of nontrivial integrable systems did not exert any real influence on the development of physics