5,322 research outputs found
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
Reductions of the Volterra and Toda chains
The Volterra and Toda chains equations are considered. A class of special
reductions for these equations are derived.Comment: LaTeX, 6 page
Gopakumar-Vafa invariants via vanishing cycles
In this paper, we propose an ansatz for defining Gopakumar-Vafa invariants of
Calabi-Yau threefolds, using perverse sheaves of vanishing cycles. Our proposal
is a modification of a recent approach of Kiem-Li, which is itself based on
earlier ideas of Hosono-Saito-Takahashi. We conjecture that these invariants
are equivalent to other curve-counting theories such as Gromov-Witten theory
and Pandharipande-Thomas theory. Our main theorem is that, for local surfaces,
our invariants agree with PT invariants for irreducible one-cycles. We also
give a counter-example to the Kiem-Li conjectures, where our invariants match
the predicted answer. Finally, we give examples where our invariant matches the
expected answer in cases where the cycle is non-reduced, non-planar, or
non-primitive.Comment: 63 pages, many improvements of the exposition following referee
comments, final version to appear in Inventione
Ultradiscretization of the solution of periodic Toda equation
A periodic box-ball system (pBBS) is obtained by ultradiscretizing the
periodic discrete Toda equation (pd Toda eq.). We show the relation between a
Young diagram of the pBBS and a spectral curve of the pd Toda eq.. The formula
for the fundamental cycle of the pBBS is obtained as a colloraly.Comment: 41 pages; 7 figure
Dynamics of Energy Transport in a Toda Ring
We present results on the relationships between persistent currents and the
known conservation laws in the classical Toda ring. We also show that
perturbing the integrability leads to a decay of the currents at long times,
with a time scale that is determined by the perturbing parameter. We summarize
several known results concerning the Toda ring in 1-dimension, and present new
results relating to the frequency, average kinetic and potential energy, and
mean square displacement in the cnoidal waves, as functions of the wave vector
and a parameter that determines the non linearity.Comment: 34 pages, 11 figures. Small changes made in response to referee's
comment
Chaos and Noise in a Truncated Toda Potential
Results are reported from a numerical investigation of orbits in a truncated
Toda potential which is perturbed by weak friction and noise. Two significant
conclusions are shown to emerge: (1) Despite other nontrivial behaviour,
configuration, velocity, and energy space moments associated with these
perturbations exhibit a simple scaling in the amplitude of the friction and
noise. (2) Even very weak friction and noise can induce an extrinsic diffusion
through cantori on a time scale much shorter than that associated with
intrinsic diffusion in the unperturbed system.Comment: 10 pages uuencoded PostScript (figures included), (A trivial
mathematical error leading to an erroneous conclusion is corrected
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
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
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
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