4,705 research outputs found
On the Equivalence of Different Lax Pairs for the Kac-van Moerbeke Hierarchy
We give a simple algebraic proof that the two different Lax pairs for the
Kac-van Moerbeke hierarchy, constructed from Jacobi respectively
super-symmetric Dirac-type difference operators, give rise to the same
hierarchy of evolution equations. As a byproduct we obtain some new recursions
for computing these equations.Comment: 8 page
Toda Lattice Solutions of Differential-Difference Equations for Dissipative Systems
In a certain class of differential-difference equations for dissipative
systems, we show that hyperbolic tangent model is the only the nonlinear system
of equations which can admit some particular solutions of the Toda lattice. We
give one parameter family of exact solutions, which include as special cases
the Toda lattice solutions as well as the Whitham's solutions in the Newell's
model. Our solutions can be used to describe temporal-spatial density patterns
observed in the optimal velocity model for traffic flow.Comment: Latex, 13 pages, 1 figur
Quasi-Solitons in Dissipative Systems and Exactly Solvable Lattice Models
A system of first-order differential-difference equations with time lag
describes the formation of density waves, called as quasi-solitons for
dissipative systems in this paper. For co-moving density waves, the system
reduces to some exactly solvable lattice models. We construct a shock-wave
solution as well as one-quasi-soliton solution, and argue that there are
pseudo-conserved quantities which characterize the formation of the co-moving
waves. The simplest non-trivial one is given to discuss the presence of a
cascade phenomena in relaxation process toward the pattern formation.Comment: REVTeX, 4 pages, 1 figur
Tau function and Hirota bilinear equations for the Extended bigraded Toda Hierarchy
In this paper we generalize the Sato theory to the extended bigraded Toda
hierarchy (EBTH). We revise the definition of the Lax equations,give the Sato
equations, wave operators, Hirota bilinear identities (HBI) and show the
existence of function . Meanwhile we prove the validity of its
Fay-like identities and Hirota bilinear equations (HBEs) in terms of vertex
operators whose coefficients take values in the algebra of differential
operators. In contrast with HBEs of the usual integrable system, the current
HBEs are equations of product of operators involving and
.Comment: 29 pages, to appear Journal of Mathematical Physics(2010
Radiationless energy exchange in three-soliton collisions
We revisit the problem of the three-soliton collisions in the weakly
perturbed sine-Gordon equation and develop an effective three-particle model
allowing to explain many interesting features observed in numerical simulations
of the soliton collisions. In particular, we explain why collisions between two
kinks and one antikink are observed to be practically elastic or strongly
inelastic depending on relative initial positions of the kinks. The fact that
the three-soliton collisions become more elastic with an increase in the
collision velocity also becomes clear in the framework of the three-particle
model. The three-particle model does not involve internal modes of the kinks,
but it gives a qualitative description to all the effects observed in the
three-soliton collisions, including the fractal scattering and the existence of
short-lived three-soliton bound states. The radiationless energy exchange
between the colliding solitons in weakly perturbed integrable systems takes
place in the vicinity of the separatrix multi-soliton solutions of the
corresponding integrable equations, where even small perturbations can result
in a considerable change in the collision outcome. This conclusion is
illustrated through the use of the reduced three-particle model.Comment: 11 pages, 14 figures, submitted for publicatio
Trace Formulas in Connection with Scattering Theory for Quasi-Periodic Background
We investigate trace formulas for Jacobi operators which are trace class
perturbations of quasi-periodic finite-gap operators using Krein's spectral
shift theory. In particular we establish the conserved quantities for the
solutions of the Toda hierarchy in this class.Comment: 7 page
Quantum Signatures of The Classical Disconnection Border
A quantum Heisenberg model with anisotropic coupling and all-to-all
interaction has been analyzed using the Bose-Einstein statistics. In
Ref.\cite{jsp} the existence of a classical energy disconnection border (EDB)
in the same kind of models has been demonstrated. We address here the problem
to find quantum signatures of the EDB. An independent definition of a quantum
disconnection border, motivated by considerations strictly valid in the quantum
regime is given. We also discuss the dynamical relevance of the quantum border
with respect to quantum magnetic reversal. Contrary to the classical case the
magnetization can flip even below the EDB through Macroscopic Quantum
Tunneling. We evaluate the time scale for magnetic reversal from statistical
and spectral properties, for a small number of particles and in the
semiclassical limit.Comment: 5 pages, 5 figure
Remarks on the Collective Quantization of the SU(2) Skyrme Model
We point out the question of ordering momentum operator in the canonical
\break quantization of the SU(2) Skyrme Model. Thus, we suggest a new
definition for the momentum operator that may solve the infrared problem that
appears when we try to minimize the Quantum Hamiltonian.Comment: 8 pages, plain tex, IF/UFRJ/9
An integrable generalization of the Toda law to the square lattice
We generalize the Toda lattice (or Toda chain) equation to the square
lattice; i.e., we construct an integrable nonlinear equation, for a scalar
field taking values on the square lattice and depending on a continuous (time)
variable, characterized by an exponential law of interaction in both discrete
directions of the square lattice. We construct the Darboux-Backlund
transformations for such lattice, and the corresponding formulas describing
their superposition. We finally use these Darboux-Backlund transformations to
generate examples of explicit solutions of exponential and rational type. The
exponential solutions describe the evolution of one and two smooth
two-dimensional shock waves on the square lattice.Comment: 14 pages, 4 figures, to appear in Phys. Rev. E http://pre.aps.org
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