128 research outputs found

### A q-difference Baxter's operator for the Ablowitz-Ladik chain

We construct the Baxter's operator and the corresponding Baxter's equation
for a quantum version of the Ablowitz Ladik model. The result is achieved by
looking at the quantum analogue of the classical Backlund transformations. For
comparison we find the same result by using the well-known Bethe ansatz
technique. General results about integrable models governed by the same
r-matrix algebra will be given. The Baxter's equation comes out to be a
q-difference equation involving both the trace and the quantum determinant of
the monodromy matrix. The spectrality property of the classical Backlund
transformations gives a trace formula representing the classical analogue of
the Baxter's equation. An explicit q-integral representation of the Baxter's
operator is discussed.Comment: 16 page

### B\"acklund Transformations for the Trigonometric Gaudin Magnet

We construct a Backlund transformation for the trigonometric classical Gaudin
magnet starting from the Lax representation of the model. The Darboux dressing
matrix obtained depends just on one set of variables because of the so-called
spectrality property introduced by E. Sklyanin and V. Kuznetsov. In the end we
mention some possibly interesting open problems.Comment: contribution to the Proc. of "Integrable Systems and Quantum
Symmetries 2009", Prague, June 18-20, 200

### Continuous and Discrete (Classical) Heisenberg Spin Chain Revised

Most of the work done in the past on the integrability structure of the
Classical Heisenberg Spin Chain (CHSC) has been devoted to studying the $su(2)$
case, both at the continuous and at the discrete level. In this paper we
address the problem of constructing integrable generalized ''Spin Chains''
models, where the relevant field variable is represented by a $N\times N$
matrix whose eigenvalues are the $N^{th}$ roots of unity. To the best of our
knowledge, such an extension has never been systematically pursued. In this
paper, at first we obtain the continuous $N\times N$ generalization of the CHSC
through the reduction technique for Poisson-Nijenhuis manifolds, and exhibit
some explicit, and hopefully interesting, examples for $3\times 3$ and $4\times
4$ matrices; then, we discuss the much more difficult discrete case, where a
few partial new results are derived and a conjecture is made for the general
case.Comment: This is a contribution to the Proc. of workshop on Geometric Aspects
of Integrable Systems (July 17-19, 2006; Coimbra, Portugal), published in
SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at
http://www.emis.de/journals/SIGMA

### Ermakov-Pinney and Emden-Fowler equations: new solutions from novel B\"acklund transformations

The class of nonlinear ordinary differential equations $y^{\prime\prime}y =
F(z,y^2)$, where F is a smooth function, is studied. Various nonlinear ordinary
differential equations, whose applicative importance is well known, belong to
such a class of nonlinear ordinary differential equations. Indeed, the
Emden-Fowler equation, the Ermakov-Pinney equation and the generalized Ermakov
equations are among them. B\"acklund transformations and auto B\"acklund
transformations are constructed: these last transformations induce the
construction of a ladder of new solutions adimitted by the given differential
equations starting from a trivial solutions. Notably, the highly nonlinear
structure of this class of nonlinear ordinary differential equations implies
that numerical methods are very difficulty to apply

### B\"acklund Transformations for the Kirchhoff Top

We construct B\"acklund transformations (BTs) for the Kirchhoff top by taking
advantage of the common algebraic Poisson structure between this system and the
$sl(2)$ trigonometric Gaudin model. Our BTs are integrable maps providing an
exact time-discretization of the system, inasmuch as they preserve both its
Poisson structure and its invariants. Moreover, in some special cases we are
able to show that these maps can be explicitly integrated in terms of the
initial conditions and of the "iteration time" $n$. Encouraged by these partial
results we make the conjecture that the maps are interpolated by a specific
one-parameter family of hamiltonian flows, and present the corresponding
solution. We enclose a few pictures where the orbits of the continuous and of
the discrete flow are depicted

### Exergy dynamics of systems in thermal or concentration non-equilibrium

The paper addresses the problem of the existence and quantification of the exergy of non-equilibrium systems. Assuming that both energy and exergy are a priori concepts, the Gibbs "available energy" A is calculated for arbitrary temperature or concentration distributions across the body, with an accuracy that depends only on the information one has of the initial distribution. It is shown that A exponentially relaxes to its equilibrium value, and it is then demonstrated that its value is different from that of the non-equilibrium exergy, the difference depending on the imposed boundary conditions on the system and thus the two quantities are shown to be incommensurable. It is finally argued that all iso-energetic non-equilibrium states can be ranked in terms of their non-equilibrium exergy content, and that each point of the Gibbs plane corresponds therefore to a set of possible initial distributions, each one with its own exergy-decay history. The non-equilibrium exergy is always larger than its equilibrium counterpart and constitutes the "real" total exergy content of the system, i.e., the real maximum work extractable from the initial system. A systematic application of this paradigm may be beneficial for meaningful future applications in the fields of engineering and natural science

### Entropy Production in the Theory of Heat Conduction in Solids

The evolution of the entropy production in solids due to heat transfer is usually associated with the Prigogine's minimum entropy production principle. In this paper, we propose a critical review of the results of Prigogine and some comments on the succeeding literature. We suggest a characterization of the evolution of the entropy production of the system through the generalized Fourier modes, showing that they are the only states with a time independent entropy production. The variational approach and a Lyapunov functional of the temperature, monotonically decreasing with time, are discussed. We describe the analytic properties of the entropy production as a function of time in terms of the generalized Fourier coefficients of the system. Analytical tools are used throughout the paper and numerical examples will support the statements

### Integral representations and zeros of the Lommel function and the hypergeometric $_1F_2$ function

We give different integral representations of the Lommel function
$s_{\mu,\nu}(z)$ involving trigonometric and hypergeometric $_2F_1$ functions.
By using classical results of Polya, we give the distribution of the zeros of
$s_{\mu,\nu}(z)$ for certain regions in the plane $(\mu,\nu)$. Further, thanks
to a well known relation between the functions $s_{\mu,\nu}(z)$ and the
hypergeometric $_1F_2$ function, we describe the distribution of the zeros of
$_1F_2$ for specific values of its parameters.Comment: 15 pages, 3 figures, 1 Tabl

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