title:
Erratum: Asymptotic entanglement manipulations can be genuinely
  irreversible. [Phys. Rev. Lett. {\bf 84}, 4260 (2000)]

abstract: This is erratum of the paper [Phys. Rev. Lett. {\bf 84}, 4260 (2000)]


title: The Zeno and anti-Zeno effects on decay in dissipative quantum systems

abstract:
  We point out that the quantum Zeno effect, i.e., inhibition of spontaneous
decay by frequent measurements, is observable only in spectrally finite
reservoirs, i.e., in cavities and waveguides, using a sequence of
evolution-interrupting pulses or randomly-modulated CW fields. By contrast,
such measurements can only accelerate decay in free space.


title: Levinson theorem for Dirac particles in one dimension

abstract:
  The scattering of Dirac particles by symmetric potentials in one dimension is
studied. A Levinson theorem is established. By this theorem, the number of
bound states with even (odd) parity, $n_+$ ($n_-$), is related to the phase
shifts $\eta_+(\pm E_k)$ [$\eta_-(\pm E_k)$] of scattering states with the same
parity at zero momentum as follows: $$\eta_\pm(\mu)+\eta_\pm(-\mu)\pm{\pi\over
2}[\sin^2\eta_\pm(\mu) -\sin^2\eta_\pm(-\mu)]=n_\pm\pi.$$ The theorem is
verified by several simple examples.


title: New set of exactly solvable complex potentials giving the real energies

abstract:
  We deform the real potential of Poeschl and Teller by a shift of its
coordinate in imaginary direction. We show that the new model remains exactly
solvable. Its bound states are constructed in closed form. Wave functions are
complex and proportional to Jacobi polynomials. Some of them diverge in the
Hermitian limit. In contrast, all their energies prove real and
shift-independent. In this sense the lost Hermiticity of our family of
Hamiltonians seems replaced by their accidental PT symmetry.


title: Catalysis of entanglement manipulation for mixed states

abstract:
  We consider entanglement-assisted remote quantum state manipulation of
bi-partite mixed states. Several aspects are addressed: we present a class of
mixed states of rank two that can be transformed into another class of mixed
states under entanglement-assisted local operations with classical
communication, but for which such a transformation is impossible without
assistance. Furthermore, we demonstrate enhancement of the efficiency of
purification protocols with the help of entanglement-assisted operations.
Finally, transformations from one mixed state to mixed target states which are
sufficiently close to the source state are contrasted to similar
transformations in the pure-state case.


title: A Kochen-Specker Theorem for Imprecisely Specified Measurement

abstract:
  A recent claim that finite precision in the design of real experiments
``nullifies'' the impact of the Kochen-Specker theorem, is shown to be
unsupportable, because of the continuity of probabilities of measurement
outcomes under slight changes in the experimental configuration.


title:
Rotational Invariance, Phase Relationships and the Quantum Entanglement
  Illusion

abstract:
  Another Bell test "loophole" - imperfect rotational invariance - is explored,
and novel realist ideas on parametric down-conversion as used in recent
"quantum entanglement" experiments are presented. The usual quantum theory of
entangled systems assumes we have rotational invariance (RI), so that
coincidence rates depend on the difference only between detector settings, not
on the absolute values. Bell tests, as such, do not necessarily require RI, but
where it fails the presentation of results in the form of coincidence curves
can be grossly misleading. Even if the well-known detection loophole were
closed, the visibility of such curves would tell us nothing about the degree of
entanglement! The problem may be especially relevant to recent experiments
using "degenerate type II parametric down-conversion" sources. Logical analysis
of the results of many experiments suggests realist explanations involving some
new physics. The systems may be more nearly deterministic than quantum theory
implies. Whilst this may be to the advantage of those attempting to make use of
the so-called "Bell correlations" in computing, encryption, "teleportation"
etc., it does mean that the systems obey ordinary, not quantum, logic.


title: Quantum Teleportation with Entangled States given by Beam Splittings

abstract:
  Quantum teleportation is rigorously discussed with coherent entang led states
given by beam splittings. The mathematical scheme of beam splitti ng has been
used to study quantum communication and quantum stochastic. We d iscuss the
teleportation process by means of coherent states in this scheme for the
following two cases: (1) Delete the vacuum part from coherent states, whose
compensation provides us a perfect teleportation from Alice to Bob. (2) Use
fully realistic (physical) coherent states, which give s a non-perfect
teleportation but shows that it is exact when the average en ergy (density) of
the coherent vectors goes to infinity.


title: The Uncertainty Way of Generalization of Coherent States

abstract:
  The three ways of generalization of canonical coherent states are briefly
reviewed and compared with the emphasis laid on the (minimum) uncertainty way.
The characteristic uncertainty relations, which include the Schroedinger and
Robertson inequalities, are extended to the case of several states. It is shown
that the standard SU(1,1) and SU(2) coherent states are the unique states which
minimize the second order characteristic inequality for the three generators. A
set of states which minimize the Schroedinger inequality for the Hermitian
components of the su_q(1,1) ladder operator is also constructed. It is noted
that the characteristic uncertainty relations can be written in the alternative
complementary form.


title:
Comment on "Demonstration of the Casimir Force in the 0.6 to 6
  micrometer Range"

abstract:
  We comment on a recently published measurement of the Casimir force for
distances in the 0.6 to 6 micrometer range between two Au surfaces (Phys. Rev.
Lett. 78, 5(1997)) and the net discrepancy reported for the comparison with
theoretical predictions (Phys. Rev. Lett. 81, 5475 (1998)).


title: Programming Pulse Driven Quantum Computers

abstract:
  Arrays of weakly-coupled quantum systems can be made to compute by subjecting
them to a sequence of electromagnetic pulses of well-defined frequency and
length. Such pulsed arrays are true quantum computers: bits can be placed in
superpositions of 0 and 1, logical operations take place coherently, and
dissipation is required only for error correction. Programming such computers
is accomplished by selecting the proper sequence of pulses.


title: Teleportation of general quantum states

abstract:
  Quantum teleportation has been introduced by Benett et al. and dis cussed by
a number of authors in the framework of the singlet state. Recentl y, a
rigorous formulation of the teleportation problem of arbitrary quantum states
by means of quantum channel was given in [IOS] based on the general c hannel
theoretical formulation of the quantum gates introduced in [OW]. In t his note
we discuss a generalization of the scheme proposed in [IOS] and we give a
general method to solve the teleportation problem in spaces of arbitr ary
finite dimensions.


title: Universe as quantum computer

abstract:
  This paper shows that universal quantum computers possess decoherent
histories in which complex adaptive systems evolve with high probability.


title: Separability of entangled q-bit pairs

abstract:
  The state of an entangled q-bit pair is specified by 15 numerical parameters
that are naturally regarded as the components of two 3-vectors and a
$3\times3$-dyadic. There are easy-to-use criteria to check whether a given pair
of 3-vectors plus a dyadic specify a 2-q-bit state; and if they do, whether the
state is entangled; and if it is, whether it is a separable state. Some
progress has been made in the search for analytical expressions for the degree
of separability. We report, in particular, the answer in the case of vanishing
3-vectors.


title: Higher order conductivity corrections to the Casimir force

abstract:
  The finite conductivity corrections to the Casimir force in two
configurations are calculated in the third and fourth orders in relative
penetration depth of electromagnetic zero oscillations into the metal. The
obtained analytical perturbation results are compared with recent computations.
Applications to the modern experiments are discussed.


title: Topological Chern indices in molecular spectra

abstract:
  Topological Chern indices are related to the number of rotational states in
each molecular vibrational band. Modification of the indices is associated to
the appearance of ``band degeneracies'', and exchange of rotational states
between two consecutive bands. The topological dynamical origin of these
indices is proven through a semi-classical approach, and their values are
computed in two examples. The relation with the integer quantum Hall effect is
briefly discussed.


title: Nonlinear Matter Wave Dynamics with a Chaotic Potential

abstract:
  We consider the case of a cubic nonlinear Schr\"{o}dinger equation with an
additional chaotic potential, in the sense that such a potential produces
chaotic dynamics in classical mechanics. We derive and describe an appropriate
semiclassical limit to such a nonlinear Schr\"{o}dinger equation, using a
semiclassical interpretation of the Wigner function, and relate this to the
hydrodynamic limit of the Gross-Pitaevskii equation used in the context of
Bose-Einstein condensation. We investigate a specific example of a
Gross-Pitaevskii equation with such a chaotic potential: the one-dimensional
delta-kicked harmonic oscillator, and its semiclassical limit. We explore the
feasibility of experimental realization of such a system in a Bose-Einstein
condensate experiment, giving a concrete proposal of how to implement such a
configuration, and considering the problem of condensate depletion.


title: Optical Holonomic Quantum Computer

abstract:
  In this paper the idea of holonomic quantum computation is realized within
quantum optics. In a non-linear Kerr medium the degenerate states of laser
beams are interpreted as qubits. Displacing devices, squeezing devices and
interferometers provide the classical control parameter space where the
adiabatic loops are performed. This results into logical gates acting on the
states of the combined degenerate subspaces of the lasers, producing any one
qubit rotations and interactions between any two qubits. Issues such as
universality, complexity and scalability are addressed and several steps are
taken towards the physical implementation of this model.


title: Coherent states for a particle on a sphere

abstract:
  The coherent states for a particle on a sphere are introduced. These states
are labelled by points of the classical phase space, that is the position on
the sphere and the angular momentum of a particle. As with the coherent states
for a particle on a circle discussed in Kowalski K {\em et al} 1996 {\em J.
Phys. A} {\bf 29} 4149, we deal with a deformation of the classical phase space
related with quantum fluctuations. The expectation values of the position and
the angular momentum in the coherent states are regarded as the best possible
approximation of the classical phase space. The correctness of the introduced
coherent states is illustrated by an example of the rotator.


title: Quantum noise in optical fibers I: stochastic equations

abstract:
  We analyze the quantum dynamics of radiation propagating in a single mode
optical fiber with dispersion, nonlinearity, and Raman coupling to thermal
phonons. We start from a fundamental Hamiltonian that includes the principal
known nonlinear effects and quantum noise sources, including linear gain and
loss. Both Markovian and frequency-dependent, non-Markovian reservoirs are
treated. This allows quantum Langevin equations to be calculated, which have a
classical form except for additional quantum noise terms. In practical
calculations, it is more useful to transform to Wigner or +$P$
quasi-probability operator representations. These result in stochastic
equations that can be analyzed using perturbation theory or exact numerical
techniques. The results have applications to fiber optics communications,
networking, and sensor technology.


title:
Quantum noise in optical fibers II: Raman jitter in soliton
  communications

abstract:
  The dynamics of a soliton propagating in a single-mode optical fiber with
gain, loss, and Raman coupling to thermal phonons is analyzed. Using both
soliton perturbation theory and exact numerical techniques, we predict that
intrinsic thermal quantum noise from the phonon reservoirs is a larger source
of jitter and other perturbations than the gain-related Gordon-Haus noise, for
short pulses, assuming typical fiber parameters. The size of the Raman timing
jitter is evaluated for both bright and dark (topological) solitons, and is
larger for bright solitons. Because Raman thermal quantum noise is a nonlinear,
multiplicative noise source, these effects are stronger for the more intense
pulses needed to propagate as solitons in the short-pulse regime. Thus Raman
noise may place additional limitations on fiber-optical communications and
networking using ultrafast (subpicosecond) pulses.


title:
Bose Condensates with 1/r Interatomic Attraction: Electromagnetically
  Induced ``Gravity''

abstract:
  We show that particular configurations of intense off-resonant laser beams
can give rise to an attractive 1/r interatomic potential between atoms located
well within the laser wavelength. Such a ``gravitational-like'' interaction is
shown to give stable Bose condensates that are self-bound (without an
additional trap) with unique scaling properties and measurably distinct
signatures.


title: Local environment can enhance fidelity of quantum teleportation

abstract:
  We show how an interaction with the environment can enhance fidelity of
quantum teleportation. To this end, we present examples of states which cannot
be made useful for teleportation by any local unitary transformations;
nevertheless, after being subjected to a dissipative interaction with the local
environment, the states allow for teleportation with genuinely quantum
fidelity. The surprising fact here is that the necessary interaction does not
require any intelligent action from the parties sharing the states. In passing,
we produce some general results regarding optimization of teleportation
fidelity by local action. We show that bistochastic processes cannot improve
fidelity of two-qubit states. We also show that in order to have their fidelity
improvable by a local process, the bipartite states must violate the so-called
reduction criterion of separability.


title: The Space-time Origin of Quantum Mechanics: Covering Law

abstract:
  Lorentz covariance imposed upon a quantum logic of local propositions for
which all observers can consistently maintain state collapse descriptions,
implies a condition on space-like separated propositions that if imposed on
generally commuting ones would lead to the covering law, and hence to a
hilbert-space model for the logic. Such a generalization can be argued if state
preparation can be conditioned to space-like separated events using EPR-type
correlations. This suggests that the covering law is related to space-time
structure, though a final understanding of it, through a self-consistency
requirement, will probably require quantum space-time.


title: Quantum Computing, NP-complete Problems and Chaotic Dynamics

abstract:
  An approach to the solution of NP-complete problems based on quantum
computing and chaotic dynamics is proposed. We consider the satisfiability
problem and argue that the problem, in principle, can be solved in polynomial
time if we combine the quantum computer with the chaotic dynamics amplifier
based on the logistic map. We discuss a possible implementation of such a
chaotic quantum computation by using the atomic quantum computer with quantum
gates described by the Hartree-Fock equations. In this case, in principle, one
can build not only standard linear quantum gates but also nonlinear gates and
moreover they obey to Fermi statistics. This new type of entaglement related
with Fermi statistics can be interesting also for quantum communication theory.

categories: quant-ph chao-dyn cond-mat.mes-hall cs.CC nlin.CD physics.atom-ph


title: Observation of radiation pressure exerted by evanescent waves

abstract:
  We report a direct observation of radiation pressure, exerted on cold
rubidium atoms while bouncing on an evanescent-wave atom mirror. We analyze the
radiation pressure by imaging the motion of the atoms after the bounce. The
number of absorbed photons is measured for laser detunings ranging from {190
MHz} to {1.4 GHz} and for angles from {0.9 mrad} to {24 mrad} above the
critical angle of total internal reflection. Depending on these settings, we
find velocity changes parallel with the mirror surface, ranging from 1 to {18
cm/s}. This corresponds to 2 to 31 photon recoils per atom. These results are
independent of the evanescent-wave optical power.


title: Field quantization by means of a single harmonic oscillator

abstract:
  A new scheme of field quantization is proposed. Instead of associating with
different frequencies different oscillators we begin with a single oscillator
that can exist in a superposition of different frequencies. The idea is applied
to the electromagnetic radiation field. Using the standard Dirac-type
mode-quantization of the electromagnetic field we obtain several standard
properties such as coherent states or spontaneous and stimulated emission. As
opposed to the standard approach the vacuum energy is finite and does not have
to be removed by any ad hoc procedure.


title: Stationary quantum source coding

abstract:
  In this paper the quantum source coding theorem is obtained for a completely
ergodic source. This results extends Shannon's classical theorem as well as
Schumacher's quantum noiseless coding theorem for memoryless sources. The
control of the memory effects requires earlier results of Hiai and Petz on high
probability subspaces.


title: Automatic Quantum Error Correction

abstract:
  Criteria are given by which dissipative evolution can transfer populations
and coherences between quantum subspaces, without a loss of coherence. This
results in a form of quantum error correction that is implemented by the joint
evolution of a system and a cold bath. It requires no external intervention
and, in principal, no ancilla. An example of a system that protects a qubit
against spin-flip errors is proposed. It consists of three spin 1/2 magnetic
particles and three modes of a resonator. The qubit is the triple quantum
coherence of the spins, and the photons act as ancilla.


title:
Entangled state quantum cryptography: Eavesdropping on the Ekert
  protocol

abstract:
  Using polarization-entangled photons from spontaneous parametric
downconversion, we have implemented Ekert's quantum cryptography protocol. The
near-perfect correlations of the photons allow the sharing of a secret key
between two parties. The presence of an eavesdropper is continually checked by
measuring Bell's inequalities. We investigated several possible eavesdropper
strategies, including pseudo-quantum non-demolition measurements. In all cases,
the eavesdropper's presence was readily apparent. We discuss a procedure to
increase her detectability.


title: Atom Chips

abstract:
  Atoms can be trapped and guided using nano-fabricated wires on surfaces,
achieving the scales required by quantum information proposals. These Atom
Chips form the basis for robust and widespread applications of cold atoms
ranging from atom optics to fundamental questions in mesoscopic physics, and
possibly quantum information systems.


title: Quantum feedback control and classical control theory

abstract:
  We introduce and discuss the problem of quantum feedback control in the
context of established formulations of classical control theory, examining
conceptual analogies and essential differences. We describe the application of
state-observer based control laws, familiar in classical control theory, to
quantum systems and apply our methods to the particular case of switching the
state of a particle in a double-well potential.


title:
Complementarity between Position and Momentum as a Consequence of
  Kochen-Specker Arguments

abstract:
  We give two simple Kochen-Specker arguments for complementary between the
position and momentum components of spinless particles, arguments that are
identical in structure to those given by Peres and Mermin for spin-1/2
particles.


title: Time of arrival through interacting environments: Tunneling processes

abstract:
  We discuss the propagation of wave packets through interacting environments.
Such environments generally modify the dispersion relation or shape of the wave
function. To study such effects in detail, we define the distribution function
P_{X}(T), which describes the arrival time T of a packet at a detector located
at point X. We calculate P_{X}(T) for wave packets traveling through a
tunneling barrier and find that our results actually explain recent
experiments. We compare our results with Nelson's stochastic interpretation of
quantum mechanics and resolve a paradox previously apparent in Nelson's
viewpoint about the tunneling time.


title:
Experiments on Sonoluminescence: Possible Nuclear and QED Aspects and
  Optical Applications

abstract:
  Experiments aimed at testing some hypothesis about the nature of Single
Bubble Sonoluminescence are discussed. A possibility to search for micro-traces
of thermonuclear neutrons is analyzed, with the aid of original low-background
neutron counter operating under conditions of the deep shielding from Cosmic
and other sources of background. Besides, some signatures of QED-contribution
to the light emission in SBSL are under the consideration, as well as new
approaches to probe a temperature inside the bubble. An applied-physics portion
of the program is presented also, in which an attention is being paid to
single- and a few-pulse light sources on the basis of SBSL.


title: Thermostatistics of q-deformed boson gas

abstract:
  We show that a natural realization of the thermostatistics of q-bosons can be
built on the formalism of q-calculus and that the entire structure of
thermodynamics is preserved if we use an appropriate Jackson derivative in
place of the ordinary thermodynamics derivative. This framework allows us to
obtain a generalized q-boson entropy which depends on the q-basic number. We
study the ideal q-boson gas in the thermodynamic limit which is shown to
exhibit Bose-Einstein condensation with higher critical temperature and
discontinuous specific heat.


title:
The Double Quantum Dot Feline Cousin of Schroedinger's Cat: An
  Experimental Testbed for a Discourse of Quantum Measurement Dichotomies

abstract:
  Intriguing dichotomies in quantum measurement theory involving the role of
the obersever, objective reality, collapse of wavefunctions and actualization
of a measurement outcome are cast into a patholigical gedanken experiment
involving a single electron in a double quantum dot system coupled via a weak
link.


title:
Coherent information analysis of quantum channels in simple quantum
  systems

abstract:
  The coherent information concept is used to analyze a variety of simple
quantum systems. Coherent information was calculated for the information decay
in a two-level atom in the presence of an external resonant field, for the
information exchange between two coupled two-level atoms, and for the
information transfer from a two-level atom to another atom and to a photon
field. The coherent information is shown to be equal to zero for all
full-measurement procedures, but it completely retains its original value for
quantum duplication. Transmission of information from one open subsystem to
another one in the entire closed system is analyzed to learn quantum
information about the forbidden atomic transition via a dipole active
transition of the same atom. It is argued that coherent information can be used
effectively to quantify the information channels in physical systems where
quantum coherence plays an important role.


title: "Partial" Fidelities

abstract:
  For pairs, omega, rho, of density operators on a finite dimensional Hilbert
space of dimension d I call k-fidelity the d - k smallest eigenvalues of |
omega^1/2 rho^1/2 |. k-fidelities are jointly concave in omega, rho. This
follows by representing them as infima over linear functions. For k = 0 known
properties of fidelity and transition probability are reproduced. Partial
fidelities characterize equivalence classes which are partially ordered in a
natural way.


title: The $\delta$-deformation of the Fock space

abstract:
  A deformation of the Fock space based on the finite difference replacement
for the derivative is introduced. The deformation parameter is related to the
dimension of the finite analogue of the Fock space.


title: Complete Separability and Fourier representations of n-qubit states

abstract:
  Necessary conditions for separability are most easily expressed in the
computational basis, while sufficient conditions are most conveniently
expressed in the spin basis. We use the Hadamard matrix to define the
relationship between these two bases and to emphasize its interpretation as a
Fourier transform. We then prove a general sufficient condition for complete
separability in terms of the spin coefficients and give necessary and
sufficient conditions for the complete separability of a class of generalized
Werner densities. As a further application of the theory, we give necessary and
sufficient conditions for full separability for a particular set of $n$-qubit
states whose densities all satisfy the Peres condition.


title: Quantum Cryptography with Entangled Photons

abstract:
  By realizing a quantum cryptography system based on polarization entangled
photon pairs we establish highly secure keys, because a single photon source is
approximated and the inherent randomness of quantum measurements is exploited.
We implement a novel key distribution scheme using Wigner's inequality to test
the security of the quantum channel, and, alternatively, realize a variant of
the BB84 protocol. Our system has two completely independent users separated by
360 m, and generates raw keys at rates of 400 - 800 bits/second with bit error
rates arround 3 percent.


title: A Fast and Compact Quantum Random Number Generator

abstract:
  We present the realization of a physical quantum random number generator
based on the process of splitting a beam of photons on a beam splitter, a
quantum mechanical source of true randomness. By utilizing either a beam
splitter or a polarizing beam splitter, single photon detectors and high speed
electronics the presented devices are capable of generating a binary random
signal with an autocorrelation time of 11.8 ns and a continuous stream of
random numbers at a rate of 1 Mbit/s. The randomness of the generated signals
and numbers is shown by running a series of tests upon data samples. The
devices described in this paper are built into compact housings and are simple
to operate.


title: Nuclear Teleportation

abstract:
  Until recently, only science-fiction authors ventured to use a term
teleportation. However, in the last few years, on the eve of upcoming new
millennium, the situation changed very much. The present report gives a
synopsis of main concepts in this area. The readers will be able to make sure
that paradoxical phenomena in the microcosm give a possibility to demonstrate
the exchange of properties between microobjects, removed at a very large
distance from each other, when no forces act between them. A new experimental
scheme with hydrogen and helium nuclei is proposed. It is expected that the
results of these experiments will be considered as teleportation of nuclear
properties of atoms of the simplest chemical elements. A problem of
teleportation of the more palpable cargo is left to the physics of the more
distant future.


title:
Life-time and hierarchy of memory in the dissipative quantum model of
  brain

abstract:
  Some recent developments of the dissipative quantum model of brain are
reported. In particular, the time-dependent frequency case is considered with
its implications on the different life-times of the collective modes.

categories: quant-ph cond-mat nlin.AO physics.bio-ph q-bio


title:
Simple pulse sequence for quantum logic operations using photon-exchange
  interactions

abstract:
  This paper has been withdrawn by the authors due to an error first noted by
M. Lukin.


title: Optimal signal ensembles

abstract:
  Classical messages can be sent via a noisy quantum channel in various ways,
corresponding to various choices of signal states of the channel. Previous work
by Holevo and by Schumacher and Westmoreland relates the capacity of the
channel to the properties of the signal ensemble. Here we describe some
properties characterizing the ensemble that maximizes the capacity, using the
relative entropy "distance" between density operators to give the results a
geometric flavor.


title: Multilevel Quantum Particle as a Few Virtual Qubits Materialization

abstract:
  A conception of virtual quantum information bit - virtual qubit - is
introduced. It is shown by means of virtual qubit representation that four
states of a single quantum particle is enough for implementation of full set of
the gates, which is necessary for creation an arbitrary algorithm for a quantum
computer. The physical nature and mutual disposition of four working states is
of no significance, if there are suitable selection rules for the particle
interaction with the external electromagnetic field pulses.


title: q-Discrete Toda Molecule Equation

abstract:
  A q-discrete version of the two-dimensional Toda molecule equation is
proposed through the direct method. Its solution, B\"acklund transformation and
Lax pair are discussed. The reduction to the q-discrete cylindrical Toda
molecule equation is also discussed.


title: Casorati Determinant Solution for the Relativistic Toda Lattice Equation

abstract:
  The relativistic Toda lattice equation is decomposed into three Toda systems,
the Toda lattice itself, B\"acklund transformation of Toda lattice and discrete
time Toda lattice. It is shown that the solutions of the equation are given in
terms of the Casorati determinant. By using the Casoratian technique, the
bilinear equations of Toda systems are reduced to the Laplace expansion form
for determinants. The $N$-soliton solution is explicitly constructed in the
form of the Casorati determinant.


title: Integrable four-dimensional symplectic maps of standard type

abstract:
  We search for rational, four-dimensional maps of standard type (x_{n+1} -
2x_n + x_{n-1} = eps f(x,eps)) possessing one or two polynomial integrals.
There are no non-trivial maps corresponding to cubic oscillators, but we find a
four-parameter family of such maps corresponding to quartic oscillators. This
seems to be the only such example.


title: The Super W_3 Conformal Algebra and the Boussinesq Hierarchy

abstract:
  The bihamiltonian structure of the N=2 Supersymmetric Boussinesq equation is
found. It is not reduced to the corresponding classical structure and hence it
describes the pure supersymmetric effect. For the supersymmetric Boussinesq
equation which contains the classical partner the Lax pair is given
explicitely. Thus we prove the integrability of the equation.


title: B\"acklund Transformations of MKdV and Painlev\'e Equations

abstract:
  For $N\ge 3$ there are $S_N$ and $D_N$ actions on the space of solutions of
the first nontrivial equation in the $SL(N) MKdV hierarchy, generalizing the
two $Z_2$ actions on the space of solutions of the standard MKdV equation.
These actions survive scaling reduction, and give rise to transformation groups
for certain (systems of) ODEs, including the second, fourth and fifth
Painlev\'e equations.


title:
On exact solution for some integrable nonlinear equations of the
  Schr\"odinger type

abstract:
  The outlook of a simple method to generate localized (soliton-like)
potentials of time-dependent Schrodinger type equations is given. The
conditions are discussed for the potentials to be real and nonsingular. For the
derivative Schrodinger equation also is discussed its relation to the
Ishimori-II model. Some pecular soliton solutions of nonlinear Schrodinger type
equations are given and discussed.


title: Lattice Virasoro from Lattice Kac-Moody

abstract:
  A new version of quantum Miura transformation on the lattice is proposed,
based on the lattice Kac-Moody algebra. A possibility of existence of the
Sugawara construction on the lattice is discussed.


title:
The upper triangular solutions to the three-state constant quantum
  Yang-Baxter equation

abstract:
  In this article we present all nonsingular upper triangular solutions to the
constant quantum Yang-Baxter equation
$R_{j_1j_2}^{k_1k_2}R_{k_1j_3}^{l_1k_3}R_{k_2k_3}^{l_2l_3}=
R_{j_2j_3}^{k_2k_3}R_{j_1k_3}^{k_1l_3}R_{k_1k_2}^{l_1l_2}$ in the three state
case, i.e. all indices ranging from 1 to 3. The upper triangular ansatz implies
729 equations for 45 variables. Fortunately many of the equations turned out to
be simple allowing us to start breaking the problem into smaller ones. In the
end we had a total of 552 solutions, but many of them were either inherited
from two-state solutions or subcases of others. The final list contains 35
nontrivial solutions, most of them new.


title:
Symmetry Reductions and Exact Solutions of a class of Nonlinear Heat
  Equations

abstract:
  Classical and nonclassical symmetries of the nonlinear heat equation
$$u_t=u_{xx}+f(u),\eqno(1)$$ are considered. The method of differential
Gr\"obner bases is used both to find the conditions on $f(u)$ under which
symmetries other than the trivial spatial and temporal translational symmetries
exist, and to solve the determining equations for the infinitesimals. A
catalogue of symmetry reductions is given including some new reductions for the
linear heat equation and a catalogue of exact solutions of (1) for cubic $f(u)$
in terms of the roots of $f(u)=0$.


title: A note on the motion of surfaces

abstract:
  We study the motion of surfaces in an intrinsic formulation in which the
surface is described by its metric and curvature tensors. The evolution
equations for the six quantities contained in these tensors are reduced in
number in two cases: (i) for arbitrary surfaces, we use principal coordinates
to obtain two equations for the two principal curvatures, highlighting the
similarity with the equations of motion of a plane curve; and (ii) for surfaces
with spatially constant negative curvature, we use parameterization by
Tchebyshev nets to reduce to a single evolution equation. We also obtain
necessary and sufficient conditions for a surface to maintain spatially
constant negative curvature as it moves. One choice for the surface's normal
motion leads to the modified-Korteweg de Vries equation,the appearance of which
is explained by connections to the AKNS hierarchy and the motion of space
curves.


title: Solutions of the Kpi Equation with Smooth Initial Data

abstract:
  The solution $u(t,x,y)$ of the Kadomtsev--Petviashvili I (KPI) equation with
given initial data $u(0,x,y)$ belonging to the Schwartz space is considered. No
additional special constraints, usually considered in literature, as
$\int\!dx\,u(0,x,y)=0$ are required to be satisfied by the initial data. The
problem is completely solved in the framework of the spectral transform theory
and it is shown that $u(t,x,y)$ satisfies a special evolution version of the
KPI equation and that, in general, $\partial_t u(t,x,y)$ has different left and
right limits at the initial time $t=0$. The conditions of the type
$\int\!dx\,u(t,x,y)=0$, $\int\!dx\,xu_y(t,x,y)=0$ and so on (first, second,
etc. `constraints') are dynamically generated by the evolution equation for
$t\not=0$. On the other side $\int\!dx\!\!\int\!dy\,u(t,x,y)$ with prescribed
order of integrations is not necessarily equal to zero and gives a nontrivial
integral of motion.


title: Deformations of Calogero-Moser Systems

abstract:
  Recent results are surveyed pertaining to the complete integrability of some
novel n-particle models in dimension one. These models generalize the
Calogero-Moser systems related to classical root systems. Quantization leads to
difference operators instead of differential operators.


title: Casorati Determinant Solutions for the Discrete Painlev\'e-II Equation

abstract:
  We present a class of solutions to the discrete Painlev\'e-II equation for
particular values of its parameters. It is shown that these solutions can be
expressed in terms of Casorati determinants whose entries are discrete Airy
functions. The analogy between the $\tau$ function for the discrete P$_{\rm
\romanno2}$ and the that of the discrete Toda molecule equation is pointed out.


title: Linearization And Solutions Of The Discrete Painlev\'e-III Equation

abstract:
  We present particular solutions of the discrete Painlev\'e III
(d-P$\rm_{III}$) equation of rational and special function (Bessel) type. These
solutions allow us to establish a close parallel between this discrete equation
and its continuous counterpart. Moreover, we propose an alternate form for
d-P$\rm_{III}$ and confirm its integrability by explicitly deriving its Lax
pair.


title: Continuous vacua in bilinear soliton equations

abstract:
  We discuss the freedom in the background field (vacuum) on top of which the
solitons are built. If the Hirota bilinear form of a soliton equation is given
by $A(D_{\vec x})\bd GF=0,\, B(D_{\vec x})(\bd FF - \bd GG)=0$ where both $A$
and $B$ are even polynomials in their variables, then there can be a continuum
of vacua, parametrized by a vacuum angle $\phi$. The ramifications of this
freedom on the construction of one- and two-soliton solutions are discussed. We
find, e.g., that once the angle $\phi$ is fixed and we choose $u=\arctan G/F$
as the physical quantity, then there are four different solitons (or kinks)
connecting the vacuum angles $\pm\phi$, $\pm\phi\pm\Pi2$ (defined modulo
$\pi$). The most interesting result is the existence of a ``ghost'' soliton; it
goes over to the vacuum in isolation, but interacts with ``normal'' solitons by
giving them a finite phase shift.


title: Local Geometric Invariants of Integrable Evolution Equations

abstract:
  The integrable hierarchy of commuting vector fields for the localized
induction equation of 3D hydrodynamics, and its associated recursion operator,
are used to generate families of integrable evolution equations which preserve
local geometric invariants of the evolving curve or swept-out surface.


title: Algorithms for the Nonclassical Method of Symmetry Reductions

abstract:
  In this article we present first an algorithm for calculating the determining
equations associated with so-called ``nonclassical method'' of symmetry
reductions (a la Bluman and Cole) for systems of partial differentail
equations. This algorithm requires significantly less computation time than
that standardly used, and avoids many of the difficulties commonly encountered.
The proof of correctness of the algorithm is a simple application of the theory
of Grobner bases. In the second part we demonstrate some algorithms which may
be used to analyse, and often to solve, the resulting systems of overdetermined
nonlinear PDEs. We take as our principal example a generalised Boussinesq
equation, which arises in shallow water theory. Although the equation appears
to be non-integrable, we obtain an exact ``two-soliton'' solution from a
nonclassical reduction.


title: On a Shallow Water Wave Equation

abstract:
  In this paper we study a shallow water equation derivable using the
Boussinesq approximation, which includes as two special cases, one equation
discussed by Ablowitz et. al. [Stud. Appl. Math., 53 (1974) 249--315] and one
by Hirota and Satsuma [J. Phys. Soc. Japan}, 40 (1976) 611--612]. A catalogue
of classical and nonclassical symmetry reductions, and a Painleve analysis, are
given. Of particular interest are families of solutions found containing a rich
variety of qualitative behaviours. Indeed we exhibit and plot a wide variety of
solutions all of which look like a two-soliton for t>0 but differ radically for
t<0. These families arise as nonclassical symmetry reduction solutions and
solutions found using the singular manifold method. This example shows that
nonclassical symmetries and the singular manifold method do not, in general,
yield the same solution set. We also obtain symmetry reductions of the shallow
water equation solvable in terms of solutions of the first, third and fifth
Painleve equations. We give evidence that the variety of solutions found which
exhibit ``nonlinear superposition'' is not an artefact of the equation being
linearisable since the equation is solvable by inverse scattering. These
solutions have important implications with regard to the numerical analysis for
the shallow water equation we study, which would not be able to distinguish the
solutions in an initial value problem since an exponentially small change in
the initial conditions can result in completely different qualitative
behaviours.


title:
Nonlinear integrable systems related to arbitrary space-time dependence
  of the spectral transform

abstract:
  We propose a general algebraic analytic scheme for the spectral transform of
solutions of nonlinear evolution equations. This allows us to give the general
integrable evolution corresponding to an arbitrary time and space dependence of
the spectral transform (in general nonlinear and with non-analytic dispersion
relations). The main theorem is that the compatibility conditions gives always
a true nonlinear evolution because it can always be written as an identity
between polynomials in the spectral variable $k$. This general result is then
used to obtain first a method to generate a new class of solutions to the
nonlinear Schroedinger equation, and second to construct the spectral transform
theory for solving initial-boundary value problems for resonant wave-coupling
processes (like self-induced transparency in two-level media, or stimulated
Brillouin scattering of plasma waves or else stimulated Raman scattering in
nonlinear optics etc...).


title: A Novel Hierarchy of Integrable Lattices

abstract:
  In the framework of the reduction technique for Poisson-Nijenhuis structures,
we derive a new hierarchy of integrable lattice, whose continuum limit is the
AKNS hierarchy. In contrast with other differential-difference versions of the
AKNS system, our hierarchy is endowed with a canonical Poisson structure and,
moreover, it admits a vector generalisation. We also solve the associated
spectral problem and explicity contruct action-angle variables through the
r-matrix approach.


title: Non Proliferation of Preimages in Integrable Mappings

abstract:
  We present an integrability criterion for rational mappings based on two
requirements. First, that a given point should have a unique preimage under the
mapping and, second, that the spontaneously appearing singularities be confined
to a few iteration steps. We present several examples of known integrable
mappings that meet these requirements and, also, use our algorithm in order to
derive new examples of integrable mappings.


title: Integrable systems and symmetric products of curves

abstract:
  We show how there is associated to each non-constant polynomial $F(x,y)$ a
completely integrable system with polynomial invariants on $\Rd$ and on
$\C{2d}$ for each $d\geq1$; in fact the invariants are not only in involution
for one Poisson bracket, but for a large class of polynomial Poisson brackets,
indexed by the family of polynomials in two variables. We show that the complex
invariant manifolds are isomorphic to affine parts of $d$-fold symmetric
products of a deformation of the algebraic curve $F(x,y)=0$, and derive the
structure of the real invariant manifolds from it. We also exhibit Lax
equations for the hyperelliptic case (i.e., when $F(x,y)$ is of the form
$y^2+f(x)$) and we show that in this case the invariant manifolds are affine
parts of distinguished (non-linear) subvarieties of the Jacobians of the
curves. As an application the geometry of the H\'enon-Heiles hierarchy --- a
family of superimposable integrable polynomial potentials on the plane --- is
revealed and Lax equations for the hierarchy are given.


title:
Multi-Hamiltonian formulation for a class of degenerate completely
  integrable systems

abstract:
  Generalizing a construction of P. Vanhaecke, we introduce a large class of
degenerate (i.e., associated to a degenerate Poisson bracket) completely
integrable systems on (a dense subset of) the space $\R^{2d+n+1}$, called the
generalized master systems. It turns out that certain generalized master
systems (with different Poisson brackets and different Hamiltonians) determine
the same Hamiltonian vector fields (and are therefore different descriptions of
the same Hamiltonian system), and that the Poisson brackets of these systems
are compatible. Consequently, our class of generalized master systems actually
consists of a (smaller) class of completely integrable systems, and our
construction yields a multi-Hamiltonian structure for these systems. As an
application, we construct a multi-Hamiltonian structure for the so-called
master systems introduced by D. Mumford.


title: On UrKdV and UrKP

abstract:
  We present two extensions of Wilson's explanation of the Miura map from MKdV
to KdV. In the first we explain the map of Svinolupov et al from a certain
UrKdV-like equation to KdV, and in the second we explain Konopelchenko's map
from the modified KP equation to KP. In the course of the latter we introduce
an ``UrKP'' system, with an infinite dimensional symmetry, providing us with a
systematic method to construct Backlund transformations for the modified KP and
KP equations.


title: The planar filament equation

abstract:
  The planar filament equation and its relation to the modified
Korteweg-deVries equation are studied in the context of Poisson geometry. The
structure of the planar filament equation is shown to be similar to that of the
3-D localized induction equation, previously studied by the authors.


title:
Localized induction equation, Heisenberg chain, and nonlinear
  Schrodinger equation

abstract:
  The three equations named in the title are examples of infinite-dimensional
completely integrable Hamiltonian systems, and are related to each other via
simple geometric constructions. In this paper, these interrelationships are
further explained in terms of the recursion operator for the Localized
Induction Equation, and the recursion operator is seen to play a variety of
roles in key geometric variational formulas.


title: Dynamical Systems Accepting the Normal Shift

abstract:
  Newtonian dynamical systems accepting the normal shift on an arbitrary
Riemannian manifold are considered. Partial differential equations forming the
weak and additional normality conditions for them are reported.

categories: solv-int alg-geom astro-ph gr-qc hep-th math.AG nlin.SI


title:
Problem of Metrizability for the Dynamical Systems Accepting the Normal
  Shift

abstract:
  The problem of metrizability for the dynamical systems accepting the normal
shift is formulated and solved. The explicit formula for the force field of
metrizable Newtonian dynamical system $\ddot\bold r=\bold F(\bold r,\dot\bold
r)$ is found.


title: Spatial Solitons in Media with Delayed-Response Optical Nonlinearities

abstract:
  Near-soliton scanning light-beam propagation in media with both
delayed-response Kerr-type and thermal nonlinearities is analyzed. The
delayed-response part of the Kerr nonlinearity is shown to be competitive as
compared to the thermal nonlinearity, and relevant contributions to a
distortion of the soliton form and phase can be mutually compensated. This
quasi-soliton beam propagation regime keeps properties of the incli- ned
self-trapped channel.


title:
Non-Linear Evolution Equations with Non-Analytic Dispersion Relations in
  2+1 Dimensions. Bilocal Approach

abstract:
  A method is proposed of obtaining (2+1)-dimensional non- linear equations
with non-analytic dispersion relations. Bilocal formalism is shown to make it
possible to represent these equations in a form close to that for their
counterparts in 1+1 dimensions.


title:
Multilinear Operators: The Natural Extension Of Hirota's Bilinear
  Formalism

abstract:
  We introduce multilinear operators, that generalize Hirota's bilinear $D$
operator, based on the principle of gauge invariance of the $\tau$ functions.
We show that these operators can be constructed systematically using the
bilinear $D$'s as building blocks. We concentrate in particular on the
trilinear case and study the possible integrability of equations with one
dependent variable. The 5th order equation of the Lax-hierarchy as well as
Satsuma's lowest-order gauge invariant equation are shown to have simple
trilinear expressions. The formalism can be extended to an arbitrary degree of
multilinearity.


title: Localized induction equation and pseudospherical surfaces

abstract:
  We describe a close connection between the localized induction equation
hierarchy of integrable evolution equations on space curves, and surfaces of
constant negative Gauss curvature.


title: Non-integrability of the mixmaster universe

abstract:
  We comment on an analysis by Contopoulos et al. which demonstrates that the
governing six-dimensional Einstein equations for the mixmaster space-time
metric pass the ARS or reduced Painlev\'{e} test. We note that this is the case
irrespective of the value, $I$, of the generating Hamiltonian which is a
constant of motion. For $I < 0$ we find numerous closed orbits with two
unstable eigenvalues strongly indicating that there cannot exist two additional
first integrals apart from the Hamiltonian and thus that the system, at least
for this case, is very likely not integrable. In addition, we present numerical
evidence that the average Lyapunov exponent nevertheless vanishes. The model is
thus a very interesting example of a Hamiltonian dynamical system, which is
likely non-integrable yet passes the reduced Painlev\'{e} test.


title: Helicity Conservation via the Noether Theorem

abstract:
  The conservation of helicity in ideal barotropic fluids is discussed from a
group theoretical point of view. A new symmetry group is introduced i.e. the
alpha group of translations. It is proven via the Noether theorem that this
group generates helicity conservation.


title:
Complete integrability of a modified vector derivative nonlinear
  Schroedinger equation

abstract:
  Oblique propagation of magnetohydrodynamic waves in warm plasmas is described
by a modified vector derivative nonlinear Schroedinger equation, if charge
separation in Poisson's equation and the displacement current in Ampere's law
are properly taken into account. This modified equation cannot be reduced to
the standard derivative nonlinear Schroedinger equation and hence its possible
integrability and related properties need to be established afresh. Indeed, the
new equation is shown to be integrable by the existence of a bi--Hamiltonian
structure, which yields the recursion operator needed to generate an infinite
sequence of conserved densities. Some of these have been found explicitly by
symbolic computations based on the symmetry properties of the new equation.
Since the new equation includes as a special case the derivative nonlinear
Schroedinger equation, the recursion operator for the latter one is now readily
available.


title:
Invariant Integrability Criterion for the Equations of Hydrodynamical
  Type

abstract:
  Invariant integrability criterion for the equations of hydrodynamical type is
found. This criterion is written in the form of vanishing for some tensor which
is derived from the velocities matrix of hydrodynamical equations.


title: Integrable deformations of a polygon

abstract:
  The new integrable mapping with a simple geometric interpretation is
presented. This mapping arise from the nonlinear superposition principle for
the B\"acklund transformations of some vector evolution equation.


title: Integrable dynamics of a discrete curve and the Ablowitz-Ladik hierarchy

abstract:
  We show that the following elementary geometric properties of the motion of a
discrete (i.e. piecewise linear) curve select the integrable dynamics of the
Ablowitz-Ladik hierarchy of evolution equations: i) the set of points
describing the discrete curve lie on the sphere S^3, ii) the distance between
any two subsequant points does not vary in time, iii) the dynamics does not
depend explicitly on the radius of the sphere. These results generalize to a
discrete context our previous work on continuous curves.


title: Numerical investigation of iso-spectral cavities built from triangles

abstract:
  We present computational approaches as alternatives to the recent microwave
cavity experiment by S. Sridhar and A. Kudrolli (Phys. Rev. Lett. {\bf 72},
2175 (1994)) on iso-spectral cavities built from triangles. A straightforward
proof of iso-spectrality is given based on the mode matching method. Our
results show that the experiment is accurate to 0.3% for the first 25 states.
The level statistics resemble those of GOE when the integrable part of the
spectrum is removed.


title: Integrable Quantum Mappings

abstract:
  We discuss the canonical structure of a class of integrable quantum mappings,
i.e. iterative canonical transformations that can be interpreted as a discrete
dynamical system. As particular examples we consider quantum mappings
associated with the lattice analogues of the KdV and MKdV equations. These
mappings possess a non-ultralocal quantum Yang-Baxter structure leading to the
existence of commuting families of exact quantum invariants. We derive the
associated quantum Miura transformations between these mappings and the
corresponding quantum bi-Hamiltonian structure.


title: New exact solutions for the discrete fourth Painlev\'e equation

abstract:
  In this paper we derive a number of exact solutions of the discrete equation
$$x_{n+1}x_{n-1}+x_n(x_{n+1}+x_{n-1})=
{-2z_nx_n^3+(\eta-3\delta^{-2}-z_n^2)x_n^2+\mu^2\over
(x_n+z_n+\gamma)(x_n+z_n-\gamma)},\eqno(1)$$ where $z_n=n\delta$ and $\eta$,
$\delta$, $\mu$ and $\gamma$ are constants. In an appropriate limit (1) reduces
to the fourth \p\ (PIV) equation $${\d^2w\over\d z^2} = {1\over2w}\left({\d
w\over\d z}\right)^2+\tfr32w^3 + 4zw^2 + 2(z^2-\alpha)w +{\beta\over
w},\eqno(2)$$ where $\alpha$ and $\beta$ are constants and (1) is commonly
referred to as the discretised fourth Painlev\'e equation. A suitable
factorisation of (1) facilitates the identification of a number of solutions
which take the form of ratios of two polynomials in the variable $z_n$. Limits
of these solutions yield rational solutions of PIV (2). It is also known that
there exist exact solutions of PIV (2) that are expressible in terms of the
complementary error function and in this article we show that a discrete
analogue of this function can be obtained by analysis of (1).


title: Symmetry Reductions and Exact Solutions of Shallow Water Wave Equations

abstract:
  In this paper we study symmetry reductions and exact solutions of the shallow
water wave (SWW) equation $$u_{xxxt} + \alpha u_x u_{xt} + \beta u_t u_{xx} -
u_{xt} - u_{xx} = 0,\eqno(1)$$ where $\alpha$ and $\beta$ are arbitrary,
nonzero, constants, which is derivable using the so-called Boussinesq
approximation. Two special cases of this equation, or the equivalent nonlocal
equation obtained by setting $u_x=U$, have been discussed in the literature.
The case $\alpha=2\beta$ was discussed by Ablowitz, Kaup, Newell and Segur
[{\it Stud.\ Appl.\ Math.}, {\bf53} (1974) 249], who showed that this case was
solvable by inverse scattering through a second order linear problem. This case
and the case $\alpha=\beta$ were studied by Hirota and Satsuma [{\it J.\ Phys.\
Soc.\ Japan}, {\bf40} (1976) 611] using Hirota's bi-linear technique. Further
the case $\alpha=\beta$ is solvable by inverse scattering through a third order
linear problem. In this paper a catalogue of symmetry reductions is obtained
using the classical Lie method and the nonclassical method due to Bluman and
Cole [{\it J.\ Math.\ Mech.\/}, {\bf 18} (1969) 1025]. The classical Lie method
yields symmetry reductions of (1) expressible in terms of the first, third and
fifth \p\ transcendents and Weierstrass elliptic functions. The nonclassical
method yields a plethora of exact solutions of (1) with $\alpha=\beta$ which
possess a rich variety of qualitative behaviours. These solutions all like a
two-soliton solution for $t<0$ but differ radically for $t>0$ and may be viewed
as a nonlinear superposition of two solitons, one travelling to the left with
arbitrary speed and the other to the right with equal and opposite speed.


title: On the Monge-Ampere equivalent of the sine-Gordon equation

abstract:
  Surfaces of constant negative curvature in Euclidean space can be described
by either the sine-Gordon equation for the angle between asymptotic directions,
or a Monge-Ampere equation for the graph of the surface. We present the
explicit form of the correspondence between these two integrable non-linear
partial differential equations using their well-known properties in
differential geometry. We find that the cotangent of the angle between
asymptotic directions is directly related to the mean curvature of the surface.
This is a Backlund-type transformation between the sine-Gordon and Monge-Ampere
equations.


title: Darboux Transformations from Reductions of the KP Hierarchy

abstract:
  The use of effective Darboux transformations for general classes Lax pairs is
discussed. The general construction of ``binary'' Darboux transformations
preserving certain properties of the operator, such as self-adjointness, is
given. The classes of Darboux transformations found include the multicomponent
BKP and CKP reductions of the KP hierarchy.


title: Maps, PDE's and Solitary Waves

abstract:
  We describe a map-based model which reproduces many of the behaviors seen in
partial differential equations (PDE's). Like PDE's, we show that this model can
support an infinite number of stationary solutions, traveling solutions,
breathing solutions, and elastically colliding solutions. Unlike PDE's, the
model can be applied with minimal computational machinery, and few sources of
numerical error. Moreover, this model clarifies possible mechanisms by which
various coherent solutions are maintained in the face of dispersion.

categories: solv-int chao-dyn nlin.CD nlin.PS nlin.SI patt-sol


title:
Transparent Potentials at Fixed Energy in Dimension Two. Fixed-Energy
  Dispersion Relations for the Fast Decaying Potentials

abstract:
  For the two-dimensional Schr\"odinger equation $$ [- \Delta
+v(x)]\psi=E\psi,\ x\in \R^2,\ E=E_{fixed}>0 \ \ \ \ \ (*)$$ at a fixed
positive energy with a fast decaying at infinity potential $v(x)$ dispersion
relations on the scattering data are given.Under "small norm" assumption using
these dispersion relations we give (without a complete proof of sufficiency) a
characterization of scattering data for the potentials from the Schwartz class
$S=C_{\infty}^{(\infty)} (\hbox{\bf R}^2).$ For the potentials with zero
scattering amplitude at a fixed energy $\scriptstyle E_{fixed}$ (transparent
potentials) we give a complete proof of this characterization. As a consequence
we construct a family (parameterized by a function of one variable) of
two-dimensional spherically-symmetric real potentials from the Schwartz class
$S$ transparent at a given energy. For the two-dimensional case (without
assumption that the potential is small) we show that there are no nonzero real
exponentially decreasing at infinity, potentials transparent at a fixed energy.
For any dimension greater or equal 1 we prove that there are no nonzero real
potentials with zero forward scattering amplitude at an energy interval. We
show that KdV-type equations in dimension 2+1 related with the scattering
problem $(*)$ (the Novikov-Veselov equations) do not preserve, in general,
these dispersion relations starting from the second one. As a corollary these
equations do not preserve, in general , the decay rate faster then $|x|^{-3}$
for initial data from the Schwartz class.

categories: solv-int funct-an hep-th math.FA nlin.SI


title: Analytic Solutions of the Vector Burgers' Equation

abstract:
  The well-known analytical solution of Burgers' equation is extended to
curvilinear coordinate systems in three-dimensions by a method which is much
simpler and more suitable to practical applications than that previously used.
The results obtained are applied to incompressible flow with cylindrical
symmetry, and also to the decay of an initially linearly increasing wind.


title:
Boundary Value Problems For Integrable Equations Compatible With The
  Symmetry Algebra

abstract:
  Boundary value problems for integrable nonlinear partial differential
equations are considered from the symmetry point of view. Families of boundary
conditions compatible with the Harry-Dym, KdV and MKdV equations and the
Volterra chain are discussed. We also discuss the uniqueness of some of these
boundary conditions.


title:
On the Existence of Localized Excitations in Nonlinear Hamiltonian
  Lattices

abstract:
  We consider time-periodic nonlinear localized excitations (NLEs) on
one-dimensional translationally invariant Hamiltonian lattices with arbitrary
finite interaction range and arbitrary finite number of degrees of freedom per
unit cell. We analyse a mapping of the Fourier coefficients of the NLE
solution. NLEs correspond to homoclinic points in the phase space of this map.
Using dimensionality properties of separatrix manifolds of the mapping we show
the persistence of NLE solutions under perturbations of the system, provided
NLEs exist for the given system. For a class of nonintegrable Fermi-Pasta-Ulam
chains we rigorously prove the existence of NLE solutions.


title: Integrable Trilinear PDE's

abstract:
  In a recent publication we proposed an extension of Hirota's bilinear
formalism to arbitrary multilinearities. The trilinear (and higher) operators
were constructed from the requirement of gauge invariance for the nonlinear
equation. Here we concentrate on the trilinear case, and use singularity
analysis in order to single out equations that are likely to be integrable. New
PDE's are thus obtained, along with others already well-known for their
integrability and for which we obtain here the trilinear expression. To appear
in the proceedings of NEEDS'94 (11-18 September, Los Alamos)


title: Variable Coefficient Third Order KdV Type of Equations

abstract:
  We show that the integrable subclassess of a class of third order
non-autonomous equations are identical with the integrable subclassess of the
autonomous ones.


title: Limits to Extensions of Burgers Equation

abstract:
  The vector Burgers equation is extended to include pressure gradients and
gravity. It is shown that within the framework of the Cole-Hopf transformation
there are no physical solutions to this problem. This result is important
because it clearly demonstrates that any extension of Burgers equation to more
interesting physical situations is strongly limited.

categories: solv-int astro-ph chao-dyn comp-gas nlin.CD nlin.CG nlin.SI


title:
Bilinear Structure and Exact Solutions of the Discrete Painlev\'e I
  Equation

abstract:
  Bilinear structure for the discrete Painlev\'e I equation is investigated.
The solution on semi-infinite lattice is given in terms of the Casorati
determinant of discrete Airy function. Based on this fact, the discrete
Painlev\'e I equation is naturally extended to a discrete coupled system.
Corresponding matrix model is also mentioned.


title:
B\"acklund transformation and the construction of the integrable
  boundary-value problem for the equation $u_{xx}-u_{tt}=e^u-e^{-2u}$

abstract:
  B\"acklund transformation for the Bullough-Dodd-Jiber-Shabat equation
$u_{xx}-u_{tt}=e^u-e^{-2u}$ is found. The construction of integrable boundary
condition for this equation together with the algebro-geometric solutions
satisfying it are suggested.


title:
Backlund Transformations and Hierarchies of Exact Solutions for the
  Fourth Painleve Equation and their Application to Discrete Equations

abstract:
  In this paper we describe B\"acklund transformations and hierarchies of exact
solutions for the fourth Painlev\'e equation (PIV) $${\d^2 w\over\d
z^2}={1\over2w}\left(\d w\over\d z\right)^2 + {{3\over2}}w^3 + 4zw^2 +
2(z^2-\alpha)w+{\beta\over w},\eqno(1){\hbox to 16pt{\hfill}}$$ with $\alpha$,
$\beta$ constants. Specifically, a nonlinear superposition principle for PIV,
hierarchies of solutions expressible in terms of complementary error or
parabolic cylinder functions as well as rational solutions will be derived.
Included amongst these hierarchies are solutions of (1) for which
$\alpha=\pm\tfr12n$ and $\beta=-\tfr12n^2$, with $n$ an integer. These
particular forms arise in quantum gravity and also satisfy a discrete analogue
of the first Painlev\'e equation. We also obtain a number of exact solutions of
the discrete fourth Painlev\'e equation $$x_{n+1}x_{n-1}+x_n(x_{n+1}+x_{n-1})=
{-2z_nx_n^3+(\eta-3\delta^{-2}-z_n^2)x_n^2+\mu^2\over
(x_n+z_n+\gamma)(x_n+z_n-\gamma)},\eqno(2){\hbox to 16pt{\hfill}}$$}%
{\narrower\noindent\baselineskip=12pt where $z_n=n\delta$ and $\eta$, $\delta$,
$\mu$ and $\gamma$ are constants, which, in an appropriate limit, reduces to
PIV (1). A suitable factorisation of (2) facilitates the identification of a
number of solutions which take the form of ratios of two polynomials in the
variable $z_n$ and the limits of these solutions yield rational solutions of
(1).


title: Symmetries and Exact Solutions of a 2+1-dimensional Sine-Gordon System

abstract:
  We investigate the classical and nonclassical reductions of the
$2+1$-dimensional sine-Gordon system of Konopelchenko and Rogers, which is a
strong generalisation of the sine-Gordon equation. A family of solutions
obtained as a nonclassical reduction involves a decoupled sum of solutions of a
generalised, real, pumped Maxwell-Bloch system. This implies the existence of
families of solutions, all occurring as a decoupled sum, expressible in terms
of the second, third and fifth Painlev\'e transcendents, and the sine-Gordon
equation. Indeed, hierarchies of such solutions are found, and explicit
transformations connecting members of each hierarchy are given. By applying a
known B\"acklund transformation for the system to the new solutions found, we
obtain further families of exact solutions, including some which are expressed
as the argument and modulus of sums of products of Bessel functions with
arbitrary coefficients. Finally, we prove the sine-Gordon system has the
Painlev\'e property, which requires the usual test to be modified, and derive a
non-isospectral Lax pair for the generalised, real, pumped Maxwell-Bloch
system.


title: Casorati Determinant Solutions for the Discrete Painlev\'e III Equation

abstract:
  The discrete Painlev\'e III equation is investigated based on the bilinear
formalism. It is shown that it admits the solutions expressed by the Casorati
determinant whose entries are given by the discrete Bessel function. Moreover,
based on the observation that these discrete Bessel functions are transformed
to the $q$-Bessel functions by a simple variable transformation, we present a
$q$-difference analogue of the Painlev\'e III equation.


title:
Period preserving nonisospectral flows and the moduli space of periodic
  solutions of soliton equations

abstract:
  Flows on the moduli space of the algebraic Riemann surfaces, preserving the
periods of the corresponding solutions of the soliton equations are studied. We
show that these flows are gradient with respect to some indefinite symmetric
flat metric arising in the Hamiltonian theory of the Whitham equations. The
functions generating these flows are conserved quantities for all the equations
simultaneously. We show that for 1+1 systems these flows can be imbedded in a
larger system of ordinary nonlinear differential equations with a rational
right-hand side. Finally these flows are used to give a complete description of
the moduli space of algebraic Riemann surfaces corresponding to periodic
solutions of the nonlinear Schr\"odinger equation.


title: Integrable systems and Riemann surfaces of infinite genus

abstract:
  To the spectral curves of smooth periodic solutions of the $n$-wave equation
the points with infinite energy are added. The resulting spaces are considered
as generalized Riemann surfcae. In general the genus is equal to infinity,
nethertheless these Riemann surfaces are similar to compact Riemann surfaces.
After proving a Riemann Roch Theorem we can carry over most of the
constructions of the finite gap potentials to all smooth periodic potentials.
The symplectic form turns out to be closely related to Serre duality. Finally
we prove that all non-linear PDE's, which belong to the focussing case of the
non-linear Schr\"odinger equation, have global solutions for arbitrary smooth
periodic inital potantials.


title: Bilinear Discrete Painleve-II and its Particular Solutions

abstract:
  By analogy to the continuous Painlev\'e II equation, we present particular
solutions of the discrete Painlev\'e II (d-P$\rm_{II}$) equation. These
solutions are of rational and special function (Airy) type. Our analysis is
based on the bilinear formalism that allows us to obtain the $\tau$ function
for d-P$\rm_{II}$. Two different forms of bilinear d-P$\rm_{II}$ are obtained
and we show that they can be related by a simple gauge transformation.


title: The Integrable Dynamics of Discrete and Continuous Curves

abstract:
  We show that the following geometric properties of the motion of discrete and
continuous curves select integrable dynamics: i) the motion of the curve takes
place in the N dimensional sphere of radius R, ii) the curve does not stretch
during the motion, iii) the equations of the dynamics do not depend explicitly
on the radius of the sphere. Well known examples of integrable evolution
equations, like the nonlinear Schroedinger and the sine-Gordon equations, as
well as their discrete analogues, are derived in this general framework.


title: String equation--2. Physical solution

abstract:
  This paper is a continuation of the paper by S.P.Novikov in Funct.Anal.Appl.,
v.24(1990), No 4, pp 196-206. String equation is by definition the equation
$[L,A]=1$ for the coefficients of two linear ordinary differential operators
$L$ and $A$. For the ``double scaling limit'' of the matrix model we always
have $L=-\partial_x^2+u(x)$, $A$ is some differential operator of the odd order
$2k+1$. In the first nontrivial case $k=1$ we have the Painelev\'e-1 (P-1)
equation. Only special real ``separatrix'' solutions of P-1 are important in
the quantum field theory. By the conjecture of Novikov these ``physical''
solutions, which are analytically exceptional probably have much stronger
symmetry then the other solutions but it is not proved until now. Two
asymptotic methods were developed in the previous paper -- nonlinear
semiclassics (or the Bogolubov-Whitham averaging method) and the linear
semiclassics for the ``Isomonodromic'' method. The nonlinear semiclassics gives
a good approximation for the general (``non-physical'') solutions of P-1 but
fails in the ``physical'' case. In our paper the linear semiclasics for the
``physical'' solutions of the P-1 equations is studied. In particular
connection between the semiclassics on Riemann surfaces and Hamiltonian
foliations on these surfaces is established.


title:
Dual Field Approach to Correlation Functions in the Heisenberg Xxz Spin
  Chain

abstract:
  We study zero temperature correlation functions of the spin-$1\over 2$
Heisenberg XXZ model in the critical regime $-1< \Delta\leq 1$ in a magnetic
field by means of the {\tenit Dual Field Approach}. We show for one particular
example how to derive determinant representations for correlation functions and
how to use these to embed the correlation functions in integrable systems of
integro-difference equations (IDE). These IDE are associated with a
Riemann-Hilbert problem.


title:
Conservation Laws in Field Dynamics or Why Boundary Motion is Exactly
  Integrable?

abstract:
  An infinite number of conserved quantities in the field dynamics $\phi_t = L
U(\phi) + \rho$ for a linear Hermitian (or anti-Hermitian) operator $L$, an
arbitrary function $U$ and a given source $\rho$ are presented. These integrals
of motion are the multipole moments of the potential created by $\phi$ in the
far-field. In the singular limit of a bistable scalar field $\phi = \phi_{\pm}$
(i.e. Ising limit) this theory describes a dissipative boundary motion (such as
Stefan or Saffman-Taylor problem that is the continuous limit of the
DLA-fractal growth) and can be exactly integrable. These conserved quantities
are the polynomial conservation laws attributed to the integrability. The
criterion for integrability is the uniqueness of the inverse potential
problem's solution.


title:
Bilinearization of a Generalized Derivative Nonlinear Schr\"odinger
  equation

abstract:
  A generalized derivative nonlinear Schr\"odinger equation,
  \ii q_t + q_{xx} + 2\ii \gamma |q|^2 q_x + 2\ii (\gamma-1)q^2 q^*_x +
(\gamma-1)(\gamma-2)|q|^4 q = 0 ,
  is studied by means of Hirota's bilinear formalism. Soliton solutions are
constructed as quotients of Wronski-type determinants. A relationship between
the bilinear structure and gauge transformation is also discussed.


title:
Nonselfintersecting magnetic orbits on the plane. Proof of Principle of
  the Overthrowing of the Cycles.

abstract:
  Beginning from 1981 one of the present authors (S.Novikov) published a series
of papers, (some of them in collaboration with I.Schmelzer and I.Taimanov)
dedicated to the development of the analog of Morse theory for the closed
1-forms -- multivalued functions and functionals -- on the finite - and
infinite-dimensional manifolds ({\bf Morse-Novikov Theory}).
  The notion of ``Multivalued action'' was understood and ``Topological
quantization of the coupling constant'' for them was formulated by Novikov in
1981 as a Corollary from the requirement, that the Feinmann Amplitude should be
one-valued on the space of fields-maps. Very beautiful analog of this theory
appeared also in the late 80-ies in the Symplectic Geometry and Topology, when
the so-called Floer Homology Theory was discovered. A very first topological
idea of this theory, formulated in early 80-ies, was the so-called ``Principle
of the Overthrowing of the Cycles''. It led to the results which were not
proved rigorously until now. Our goal is to prove some of them.
  We study the motion of a classical charged particle on the Euclidean plane in
a magnetic field orthogonal to this field. The trajectories of this motion can
be characterized as extremals of the ``Maupertui--Fermat'' functional. We show
that for any smooth everyvhere positive double periodic magnetic field for any
fixed energy there exist at least two different periodic convex extremals, such
that the value of the Maupertui-Fermat functional is positive for them. If all
such extremals are nondegenerate in the sense of Morse in the space of
nonparameterized curves then for any energy there exist at least 4 periodic
convex extremals with the Morse indices (1,2,2,3).


title:
Lattice and q-difference Darboux-Zakharov-Manakov systems via
  $\bar{\partial}$-dressing method

abstract:
  A general scheme is proposed for introduction of lattice and q-difference
variables to integrable hierarchies in frame of $\bar{\partial}$-dressing
method . Using this scheme, lattice and q-difference Darboux-Zakharov-Manakov
systems of equations are derived. Darboux, B\"acklund and Combescure
transformations and exact solutions for these systems are studied.


title: Integrable nonlinear field equations and loop algebra structures

abstract:
  We apply the (direct and inverse) prolongation method to a couple of
nonlinear Schr{\"o}dinger equations. These are taken as a laboratory field
model for analyzing the existence of a connection between the integrability
property and loop algebras. Exploiting a realization of the Kac-Moody type of
the incomplete prolongation algebra associated with the system under
consideration, we develop a procedure with allows us to generate a new class of
integrable nonlinear field equations containing the original ones as a special
case.


title: Geometric phase effects for wavepacket revivals

abstract:
  The study of wavepacket revivals is extended to the case of Hamiltonians
which are made time-dependent through the adiabatic cycling of some parameters.
It is shown that the quantal geometric phase (Berry's phase) causes the revived
packet to be displaced along the classical trajectory, by an amount equal to
the classical geometric phase (Hannay's angle), in one degree of freedom. A
physical example illustrating this effect in three degrees of freedom is
mentioned.


title: Shock waves in the dissipative Toda lattice

abstract:
  We consider the propagation of a shock wave (SW) in the damped Toda lattice.
The SW is a moving boundary between two semi-infinite lattice domains with
different densities. A steadily moving SW may exist if the damping in the
lattice is represented by an ``inner'' friction, which is a discrete analog of
the second viscosity in hydrodynamics. The problem can be considered
analytically in the continuum approximation, and the analysis produces an
explicit relation between the SW's velocity and the densities of the two
phases. Numerical simulations of the lattice equations of motion demonstrate
that a stable SW establishes if the initial velocity is directed towards the
less dense phase; in the opposite case, the wave gradually spreads out. The
numerically found equilibrium velocity of the SW turns out to be in a very good
agreement with the analytical formula even in a strongly discrete case. If the
initial velocity is essentially different from the one determined by the
densities (but has the correct sign), the velocity does not significantly
alter, but instead the SW adjusts itself to the given velocity by sending
another SW in the opposite direction.


title: Discrete Toda Field Equations

abstract:
  There are two-dimensional Toda field equations corresponding to each (finite
or affine) Lie algebra. The question addressed in this note is whether there
exist integrable discrete versions of these. It is shown that for certain
algebras (such as $A_n$, $A_n^{(1)}$ and $B_n$) there do, but some of these
systems are defined on the half-plane rather than the full two-dimensional
lattice.


title: The Hamiltonian structure of the dispersionless Toda hierarchy

abstract:
  The Hamiltonian structure of the two-dimensional dispersionless Toda
hierarchy is studied, this being a particular example of a system of
hydrodynamic type. The polynomial conservation laws for the system turn out,
after a change of variable, to be associated with the axially symmetric
solutions of the 3-dimensional Laplace equation and this enables a generating
function for the Hamiltonian densities to be derived in closed form.


title: Isothermic surfaces in $\E^3$ as soliton surfaces

abstract:
  We show that the theory of isothermic surfaces in $\E^3$ -- one of the oldest
branches of differential geometry -- can be reformulated within the modern
theory of completely integrable (soliton) systems. This enables one to study
the geometry of isothermic surfaces in $\E^3$ by means of powerful spectral
methods available in the soliton theory. Also the associated non-linear system
is interesting in itself since it displays some unconventional soliton features
and, physically, could be applied in the theory of infinitesimal deformations
of membranes.


title:
On The Observability For Distributed Systems By Means Of Linear
  Operations

abstract:
  An observability problem for linear autonomous distributed systems in the
class of linear operations is considered. A criterion of observability with
respect to terminal state has been proved. A connection with observability with
respect to initial state is discussed.


title:
Explicit Integration of the Full Symmetric Toda Hierarchy and the
  Sorting Property

abstract:
  We give an explicit formula for the solution to the initial value problem of
the full symmetric Toda hierarchy. The formula is obtained by the
orthogonalization procedure of Szeg\"{o}, and is also interpreted as a
consequence of the QR factorization method of Symes \cite{symes}. The sorting
property of the dynamics is also proved for the case of a generic symmetric
matrix in the sense described in the text, and generalizations of tridiagonal
formulae are given for the case of matrices with $2M+1$ nonzero diagonals.


title: Supersymmetric Harry Dym Type Equations

abstract:
  A supersymmetric version is proposed for the well known Harry Dym system. A
general class super Lax operator which leads to consistent equations is
considered.


title:
Theory of Pump Depletion and Spike Formation in Stimulated Raman
  Scattering

abstract:
  By using the inverse spectral transform, the SRS equations are solved and the
explicit output data is given for arbitrary laser pump and Stokes seed profiles
injected on a vacuum of optical phonons. For long duration laser pulses, this
solution is modified such as to take into account the damping rate of the
optical phonon wave. This model is used to interprete the experiments of Druhl,
Wenzel and Carlsten (Phys. Rev. Lett., (1983) vol. 51, p. 1171), in particular
the creation of a spike of (anomalous) pump radiation. The related nonlinear
Fourier spectrum does not contain discrete eigenvalue, hence this Raman spike
is not a soliton.


title:
On the dynamics of noncanonically coupled oscillators and its hidden
  superstructure

abstract:
  The classical and quantum dynamics of the noncanonically coupled oscillators
is considered. It is shown that though the classical dynamics is well--defined
for both harmonic and anharmonic oscillators, the quantum one is well--defined
in the harmonic case, admits a hidden (super)Hamiltonian formulation, and thus,
preserves the initial operator relations, whereas a na\"\i ve quantization of
the anharmonic case meets with principal difficulties.


title: Separation of Variables. New Trends.

abstract:
  The review is based on the author's papers since 1985 in which a new approach
to the separation of variables (\SoV) has being developed. It is argued that
\SoV, understood generally enough, could be the most universal tool to solve
integrable models of the classical and quantum mechanics. It is shown that the
standard construction of the action-angle variables from the poles of the
Baker-Akhiezer function can be interpreted as a variant of \SoV, and moreover,
for many particular models it has a direct quantum counterpart. The list of the
models discussed includes XXX and XYZ magnets, Gaudin model, Nonlinear
Schr\"odinger equation, $SL(3)$-invariant magnetic chain. New results for the
3-particle quantum Calogero-Moser system are reported.


title:
$V_{(1,1)}^{(t)}$-PERTURBED MODELS OF CFT AND THEIR QUANTUM GROUP
  SYMMETRY

abstract:
  We propose a new massive integrable model in quantum field theory. This model
is obtained as a perturbed model of the minimal conformal field theories on the
hyper-elliptic surfaces by a particular relavant operator $V_{(1,1)}^{(t)}$.
The non-local conserved charges of the model and their $q$-deformed algebra are
also constructed explicitly.


title:
Painlev\'{e} Analysis and Exact Solutions of a Modified Boussinesq
  Equation

abstract:
  We consider a modified Boussinesq type equation. The Painlev\'{e} test of the
WTC method is performed for this equation and it shows that the equation has
weak Painlev\'{e} property. Some exact solutions are constructed.


title:
Topics in nonhamiltonian (magnetic-type) interaction of classical
  hamiltonian dynamical systems. I

abstract:
  A convenient algebraic structure to describe some forms of dynamics of two
hamiltonian systems with nonpotential (magnetic--type) interaction is
considered. An algebraic mechanism of generation of such dynamics is explored
on simple "toy" examples and models. Nonpotential chains and their continuum
limits are also considered. Examples of hybrid couplings with both potential
and nonpotential (magnetic--type) interactions are discussed.


title: On the algebraic non-integrability of the Halphen system

abstract:
  It is proved that the Halphen system of ordinary differential equations has
no non-trivial rational first integrals.


title: Integrable boundary conditions for the Toda lattice

abstract:
  The problem of construction of the boundary conditions for the Toda lattice
compatible with its higher symmetries is considered. It is demonstrated that
this problem is reduced to finding of the differential constraints consistent
with the ZS-AKNS hierarchy. A method of their construction is offered based on
the B\"acklund transformations. It is shown that the generalized Toda lattices
corresponding to the non-exceptional Lie algebras of finite growth can be
obtained by imposing one of the four simplest integrable boundary conditions on
the both ends of the lattice. This fact allows, in particular, to solve the
problem of reduction of the series $A$ Toda lattices into the series $D$ ones.
Deformations of the found boundary conditions are presented which leads to the
Painlev\'e type equations.
  Key words: Toda lattice, boundary conditions, integrability, B\"acklund
transformation, Lie algebras, Painlev\'e equations


title: Toda Hierarchy with Indefinite Metric

abstract:
  We consider a generalization of the full symmetric Toda hierarchy where the
matrix $\tilde {L}$ of the Lax pair is given by $\tilde {L}=LS$, with a full
symmetric matrix $L$ and a nondegenerate diagonal matrix $S$. The key feature
of the hierarchy is that the inverse scattering data includes a class of
noncompact groups of matrices, such as $O(p,q)$. We give an explicit formula
for the solution to the initial value problem of this hierarchy. The formula is
obtained by generalizing the orthogonalization procedure of Szeg\"{o}, or the
QR factorization method of Symes. 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. The $\tau$-function
structure for the tridiagonal hierarchy is also studied.


title:
Approximate Lie Group Analysis of a Model Advection Equation on an
  Unstructured Grid

abstract:
  A technique of ``approximate group analysis'' recently developed by Baikov,
Gazizov and Ibragimov is applied to a differential approximation (otherwise
referred to as an equivalent differential equation) corresponding to the finite
difference approximation of a nonlinear advection equation on unstructured
grid. We determine which groups from the infinite variety of groups admitted by
a nonlinear advection equation ``survive'' the discretization. The situations
arising for different choices of an arbitrary function (local speed of
propagation) are also studied.


title:
Conditional Lie-B\"acklund symmetry and reduction of evolution
  equations.

abstract:
  We suggest a generalization of the notion of invariance of a given partial
differential equation with respect to Lie-B\"acklund vector field. Such
generalization proves to be effective and enables us to construct principally
new Ans\"atze reducing evolution-type equations to several ordinary
differential equations. In the framework of the said generalization we obtain
principally new reductions of a number of nonlinear heat conductivity equations
$u_t=u_{xx}+F(u,u_x)$ with poor Lie symmetry and obtain their exact solutions.
It is shown that these solutions can not be constructed by means of the
symmetry reduction procedure.


title: Separation of Variables in BC-type Gaudin Magnet

abstract:
  The integrable system is introduced based on the Poisson $ rs $-matrix
structure. This is a generalization of the Gaudin magnet, and in SL(2) case
isomorphic to the generalized Neumann model. The separation of variables is
discussed for both classical and quantum case.


title:
Infinite-Dimensional Estabrook-Wahlquist Prolongations for the
  sine-Gordon Equation

abstract:
  We are looking for the universal covering algebra for all symmetries of a
given pde, using the sine-Gordon equation as a typical example for a
non-evolution equation. For non-evolution equations, Estabrook-Wahlquist
prolongation structures for non-local symmetries depend on the choice of a
specific sub-ideal, of the contact module, to define the pde. For each
inequivalent such choice we determine the most general solution of the
prolongation equations, as sub-algebras of the (infinite-dimensional) algebra
of all vector fields over the space of non-local variables associated with the
pde, in the style of Vinogradov covering spaces. We show explicitly how
previously-known prolongation structures, known to lie within the Kac-Moody
algebra, $A_1^{(1)}$, are special cases of these general solutions, although we
are unable to identify the most general solutions with previously-studied
algebras. We show the existence of gauge transformations between prolongation
structures, viewed as determining connections over the solution space, and use
these to relate (otherwise) distinct algebras. Faithful realizations of the
universal algebra allow integral representations of the prolongation structure,
opening up interesting connections with algebras of Toeplitz operators over
Banach spaces, an area that has only begun to be explored.


title:
Algebraic integrable dynamical systems, 2+1-dimensional models in wholly
  discrete space-time, and inhomogeneous models in 2-dimensional statistical
  physics

abstract:
  This paper is devoted to constructing and studying exactly solvable dynamical
systems in discrete time obtained from some algebraic operations on matrices,
to reductions of such systems leading to classical field theory models in
2+1-dimensional wholly discrete space-time, and to connection between those
field theories and inhomogoneous models in 2-dimensional statistical physics.


title: Holomorphic Curves and Toda Systems

abstract:
  Geometry of holomorphic curves from point of view of open Toda systems is
discussed. Parametrization of curves related this way to non-exceptional simple
Lie algebras is given. This gives rise to explicit formulas for minimal
surfaces in real, complex and quaternionic projective spaces or complex
quadrics. The paper generalizes the well known connection between minimal
surfaces in $\EE^{3}$, their Weierstrass representation in terms of holomorphic
functions and the general solution to the Liouville equation.

categories: solv-int alg-geom dg-ga math.AG math.DG nlin.SI


title:
Iso-spectral deformations of general matrix and their reductions on Lie
  algebras

abstract:
  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.


title: Fredholm determinants and the mKdV/sinh-Gordon hierarchies

abstract:
  For a particular class of integral operators $K$ we show that the quantity
\[\ph:=\log \det (I+K)-\log \det (I-K)\] satisfies both the integrated mKdV
hierarchy and the sinh-Gordon hierarchy. This proves a conjecture of
Zamolodchikov.


title: Integrable Models on Hyper-Elliptic Surfaces

abstract:
  We present an elementary introduction to the construction of integrable
models on hyper-elliptic surfaces for non specialists; also, we present some of
the details of the paper `solv-int/9504002' for the more interested readers.
(Based on a talk given at the MRST 95 meeting by C. E.)


title:
Multivariable continuous Hahn and Wilson polynomials related to
  integrable difference systems

abstract:
  Multivariable generalizations of the continuous Hahn and Wilson polynomials
are introduced as eigenfunctions of rational Ruijsenaars type difference
systems with an external field.


title:
On the new approach to variable separation in the time-dependent
  Schr\"odinger equation with two space dimensions

abstract:
  We suggest an effective approach to separation of variables in the
Schr\"odinger equation with two space variables. Using it we classify
inequivalent potentials $V(x_1,x_2)$ such that the corresponding Schr\" odinger
equations admit separation of variables. Besides that, we carry out separation
of variables in the Schr\" odinger equation with the anisotropic harmonic
oscillator potential $V=k_1x_1^2+k_2x_2^2$ and obtain a complete list of
coordinate systems providing its separability. Most of these coordinate systems
depend essentially on the form of the potential and do not provide separation
of variables in the free Schr\" odinger equation ($V=0$).


title:
On the integrability of stationary and restricted flows of the KdV
  hierarchy.

abstract:
  A bi--Hamiltonian formulation for stationary flows of the KdV hierarchy is
derived in an extended phase space. A map between stationary flows and
restricted flows is constructed: in a case it connects an integrable
Henon--Heiles system and the Garnier system. Moreover a new integrability
scheme for Hamiltonian systems is proposed, holding in the standard phase
space.


title:
Henon-Heiles Hamiltonian for Coupled Upper-Hybrid and Magnetoacoutic
  Waves in Magnetized Plasmas

abstract:
  We show that the coupled mode equations for the stationary propagation of
upper--hybrid and magnetoacoustic waves in magnetized electron--ion plasmas
with negative group dispersion can be exactly derived from the generalized
\Henon--Heiles Hamiltonian. The parameter regimes for the integrable cases of
the coupled mode equations have been explicitly obtained. For positive group
dispersion of the upper--hybrid waves, the relevant governing equations lead to
a novel Hamiltonian where the kinetic energy is not positive definite.


title:
Hamiltonian Structures on Coadjoint Orbits of Semidirect Product of
  $G=Diff_+(S^{1})$ and $C^{\infty}(S^1, {\bf R})$

abstract:
  We consider the semidirect product of diffeomorphisms of the circle
$D={Diff}_+(S^1)$ and $C^{\infty}(S^{1}, {\bf R})$ functions, classify its
coadjoint orbits and prove the integrability of hamiltonian (Generalized
Dispersive Water Waves (DWW) and KdV-type) systems related to corresponding Lie
algebra centrally extended by Kac-Moody, Virasoro and semidirect product
cocycles with arbitrary coefficients.


title:
Integer spin particles necessarily produce half-integer angular momentum
  in a simple complex and periodic Hamiltonian

abstract:
  Exact wave functions are is derived from an azimuthally periodic a
self-consistent quantum Hamiltonian in 2+1 dimensions using both the
Klein-Gordon and the Schroedinger equations. It isWe shown that, curiously, for
both relativistic and non-relativistic equations, integer spin wave equations
necessarily produce half-integer angular momentum in this potential. We find
additionally that the higher energy, relativistic, solutions require an
asymptotically free potential, while the lower energy, Schroedinger, solutions
can exist in a potential that grows linearly with r. These are purely
mathematical results, however we speculate on possible physical
interpretations.


title: Separation of variables in the $A_2$ type Jack polynomials

abstract:
  An integral operator $M$ is constructed performing a separation of variables
for the 3-particle quantum Calogero-Sutherland (CS) model. Under the action of
$M$ the CS eigenfunctions (Jack polynomials for the root system $A_2$) are
transformed to the factorized form $\phi(y_1)\phi(y_2)$, where $\phi(y)$ is a
trigonometric polynomial of one variable expressed in terms of the ${}_3F_2$
hypergeometric series. The inversion of $M$ produces a new integral
representation for the $A_2$ Jack polynomials.


title: Non-Fuchsian extension to the Painlev\'e test

abstract:
  We consider meromorphic particular solutions of nonlinear ordinary
differential equations and perform a perturbation {\it \`a la} Poincar\'e
making their linearized equation non-Fuchsian at the movable pole and Fuchsian
at infinity. When the nonlinear equation possesses movable logarithms, they are
detected sooner than with the perturbative (Fuchsian) Painlev\'e test.


title:
Numerical Studies of Localized Structures on an Uneven Bottom in Two
  Dimensions

abstract:
  The Davey-Stewartson (DS) equations with a perturbation term are presented by
taking a fluid system as an example on an uneven bottom. Stability of dromions,
solutions of the DS equations with localized structures, against the
perturbation is investigated numerically. Dromions decay exponentially under an
effect of the perturbation, while they travel stably after the effect
disappears. The decay ratio of dromions is found to have relation to velocities
of dromions. The important role played by the mean flow, which acts as an
external force to the system, is discussed. These results show that dromions
are quite stable as a localized structure in two dimensions, and they are
expected to observed in various physical systems such as fluid or plasma
systems.


title: Symmetry approach in boundary value problems

abstract:
  The problem of construction of the boundary conditions for nonlinear
equations is considered compatible with their higher symmetries. Boundary
conditions for the sine-Gordon, Jiber-Shabat and KdV equations are discussed.
New examples are found for the Jiber-Shabat equation.


abstract:
  Hidden symmetry of the quantum Calogero-Moser system with the inverse-square
potential is explicitly demonstrated in algebraic sense. We find the underlying
algebra explaining the super-integrability phenomenon for this system.
Applications to related multi-variable Bessel functions are also discussed.


title: The relativistic Calogero model in an external field

abstract:
  Recent results are surveyed regarding the spectrum and eigenfunctions of the
inverse square Calogero model with harmonic confinement and its relativistic
analogue.