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At least 325 records · Page 18Linked to original sources
arXiv1997

On an integrable discretization of the modified Korteweg-de Vries equation

We find time discretizations for the two ''second flows'' of the Ablowitz-Ladik hierachy. These discretizations are described by local equations of motion, as opposed to the previously known ones, due to Taha and Ablowitz. Certain superpositions of our maps allow a one-field reduction and serve therefore as valid space-time discretizations of the modified Korteweg-de Vries equation. We expect the performance of these discretizations to be much better then that of the Taha-Ablowitz scheme. The way of finding interpolating Hamiltonians for our maps is also indicated, as well as the solution of an initial value problem in terms of matrix factorizations.

solv-int↗
arXiv1997

On the Integrable Hierarchies Associated With N=2 Super $W_n$ Algebra

A new Lax operator is proposed from the viewpoint of constructing the integrable hierarchies related with N=2 super $W_n$ algebra. It is shown that the Poisson algebra associated to the second Hamiltonian structure for the resulted hierarchy contains the N=2 super Virasoro algebra as a proper subalgebra. The simplest cases are discussed in detail. In particular, it is proved that the supersymmetric two-boson hierarchy is one of N=2 supersymmetric KdV hierarchies. Also, a Lax operator is supplied for one of N=2 supersymmetric Boussinesq hierarchies.

solv-int↗
arXiv1997

A Lax Description for Polytropic Gas Dynamics

We give a Lax description for the system of polytropic gas equations. The special structure of the Lax function naturally leads to the two infinite sets of conserved charges associated with this system. We obtain closed form expressions for the conserved charges as well as the generating functions for them. We show how the study of these generating functions can naturally lead to the recursion relation between the conserved quantities as well as the higher order Hamiltonian structures.

solv-int↗
arXiv1997

Exactly Solvable Models in Arbitrary Dimensions

We construct a new class of quasi-exactly solvable many-body Hamiltonians in arbitrary dimensions, whose ground states can have any correlations we choose. Some of the known correlations in one dimension and some recent novel correlations in two and higher dimensions are reproduced as special cases. As specific interesting examples, we also write down some new models in two and higher dimensions with novel correlations.

hep-th↗
arXiv1997

Some comments on the bi(tri)-Hamiltonian structure of Generalized AKNS and DNLS hierarchies

We give the correct prescriptions for the terms involving the inverse of the derivative of the delta function, in the Hamiltonian structures of the AKNS and DNLS systems, in order for the Jacobi identities to hold. We also establish that the sl(2) AKNS and DNLS systems are tri-Hamiltonians and construct two compatible Hamiltonian structures for the sl(3) AKNS system. We also give a derivation of the recursion operator for the sl(n+1) DNLS system.

solv-int↗
arXiv1998

Higher-dimensional Generalisations of the Euler Top Equations

Generalisations of the familiar Euler top equations in three dimensions are proposed which admit a sufficiently large number of conservation laws to permit integrability by quadratures. The usual top is a classical analogue of the Nahm equations. One of the examples discussed here is a seven-dimensional Euler top, which arises as a classical counterpart to the eight-dimensional self-dual equations which are currently believed to play a role in new developments in string theory.

hep-th↗
arXiv1997

A New ``Dual'' Symmetry Structure of the KP Hierarchy

A new infinite set of commuting additional (``ghost'') symmetries is proposed for the KP-type integrable hierarchy. These symmetries allow for a Lax representation in which they are realized as standard isospectral flows. This gives rise to a new double-KP hierarchy embedding ``ghost'' and original KP-type Lax hierarchies connected to each other via a ``duality'' mapping exchanging the isospectral and ``ghost'' ``times''. A new representation of 2D Toda lattice hierarchy as a special Darboux-Backlund orbit of the double-KP hierarchy is found and parametrized entirely in terms of (adjoint) eigenfunctions of the original KP subsystem.

solv-int↗
arXiv1998

General Solution of 7D Octonionic Top Equation

The general solution of a 7D analogue of the 3D Euler top equation is shown to be given by an integration over a Riemann surface with genus 9. The 7D model is derived from the 8D $Spin(7)$ invariant self-dual Yang-Mills equation depending only upon one variable and is regarded as a model describing self-dual membrane instantons. Several integrable reductions of the 7D top to lower target space dimensions are discussed and one of them gives 6, 5, 4D descendants and the 3D Euler top associated with Riemann surfaces with genus 6, 5, 2 and 1, respectively.

hep-th↗
arXiv1998

N Fermion Ground State of Calogero-Sutherland Type Models in Two and Higher Dimensions

I obtain the exact ground state of $N$-fermions in $D$-dimensions $(D \geq 2)$ in case the $N$ particles are interacting via long-ranged two-body and three-body interactions and further they are also interacting via the harmonic oscillator potential. I also obtain the $N$-fermion ground state in case the oscillator potential is replaced by an $N$-body Coulomb-like interaction.

cond-mat↗
arXiv1998

On a class of dynamical systems both quasi-bi-Hamiltonian and bi-Hamiltonian

It is shown that a class of dynamical systems (encompassing the one recently considered by F. Calogero [J. Math. Phys. 37 (1996) 1735]) is both quasi-bi-Hamiltonian and bi-Hamiltonian. The first formulation entails the separability of these systems; the second one is obtained trough a non canonical map whose form is directly suggested by the associated Nijenhuis tensor.

solv-int↗
arXiv1998

Extending Hamiltonian Operators to Get Bi-Hamiltonian Coupled KdV Systems

An analysis of extension of Hamiltonian operators from lower order to higher order of matrix paves a way for constructing Hamiltonian pairs which may result in hereditary operators. Based on a specific choice of Hamiltonian operators of lower order, new local bi-Hamiltonian coupled KdV systems are proposed. As a consequence of bi-Hamiltonian structure, they all possess infinitely many symmetries and infinitely many conserved densities.

solv-int↗
arXiv1998

Correlations in the impenetrable electron gas

We consider non-relativistic electrons in one dimension with infinitely strong repulsive delta function interaction. We calculate the long-time, large-distance asymptotics of field-field correlators in the gas phase. The gas phase at low temperatures is characterized by the ideal gas law. We calculate the exponential decay, the power law corrections and the constant factor of the asymptotics. Our results are valid at any temperature. They simplify at low temperatures, where they are easily recognized as products of free fermionic correlation functions with corrections arising due to the interaction.

cond-mat↗
arXiv1998

Exactly integrable family of generalized Hubbard models with twisted Yangian symmetry

A strongly correlated electron system with controlled hopping, in the line of the recently proposed generalized Hubbard models as candidates for high T_c-superconductors, is considered. The model along with a whole class of such systems are shown to be completely integrable with explicit quantum R-matrices and the Lax operators. Inspite of novelties in the Bethe ansatz solution, the final results do not deviate much from those of the standard Hubbard model. However, the symmetry of the model is changed to a recently discovered twisted Yangian symmetry.

cond-mat↗
arXiv1998

On Discretizations of the Vector Nonlinear Schrodinger Equation

Two discretizations of the vector nonlinear Schrodinger (NLS) equation are studied. One of these discretizations, referred to as the symmetric system, is a natural vector extension of the scalar integrable discrete NLS equation. The other discretization, referred to as the asymmetric system, has an associated linear scattering pair. General formulae for soliton solutions of the asymmetric system are presented. Formulae for a constrained class of solutions of the symmetric system may be obtained. Numerical studies support the hypothesis that the symmetric system has general soliton solutions.

solv-int↗