3,626 research outputs found
Transformations of Heun's equation and its integral relations
We find transformations of variables which preserve the form of the equation
for the kernels of integral relations among solutions of the Heun equation.
These transformations lead to new kernels for the Heun equation, given by
single hypergeometric functions (Lambe-Ward-type kernels) and by products of
two hypergeometric functions (Erd\'elyi-type). Such kernels, by a limiting
process, also afford new kernels for the confluent Heun equation.Comment: This version was published in J. Phys. A: Math. Theor. 44 (2011)
07520
Green's function for a Schroedinger operator and some related summation formulas
Summation formulas are obtained for products of associated Lagurre
polynomials by means of the Green's function K for the Hamiltonian H =
-{d^2\over dx^2} + x^2 + Ax^{-2}, A > 0. K is constructed by an application of
a Mercer type theorem that arises in connection with integral equations. The
new approach introduced in this paper may be useful for the construction of
wider classes of generating function.Comment: 14 page
Properties of the series solution for Painlevé I
We present some observations on the asymptotic behaviour of the coefficients in the Laurent series expansion of solutions of the first Painlevé equation. For the general solution, explicit recursive formulae for the Taylor expansion of the tau-function around a zero are given, which are natural extensions of analogous formulae for the elliptic sigma function, as given by Weierstrass. Numerical and exact results on the symmetric solution which is singular at the origin are also presented
Exact Thermodynamics of the Double sinh-Gordon Theory in 1+1-Dimensions
We study the classical thermodynamics of a 1+1-dimensional double-well
sinh-Gordon theory. Remarkably, the Schrodinger-like equation resulting from
the transfer integral method is quasi-exactly solvable at several temperatures.
This allows exact calculation of the partition function and some correlation
functions above and below the short-range order (``kink'') transition, in
striking agreement with high resolution Langevin simulations. Interesting
connections with the Landau-Ginzburg and double sine-Gordon models are also
established.Comment: 4 pages, 3 figures (embedded using epsf), uses RevTeX plus macro
(included). Minor revision to match journal version, Phys. Rev. Lett. (in
press
Occurrence of periodic Lam\'e functions at bifurcations in chaotic Hamiltonian systems
We investigate cascades of isochronous pitchfork bifurcations of
straight-line librating orbits in some two-dimensional Hamiltonian systems with
mixed phase space. We show that the new bifurcated orbits, which are
responsible for the onset of chaos, are given analytically by the periodic
solutions of the Lam\'e equation as classified in 1940 by Ince. In Hamiltonians
with C_ symmetry, they occur alternatingly as Lam\'e functions of period
2K and 4K, respectively, where 4K is the period of the Jacobi elliptic function
appearing in the Lam\'e equation. We also show that the two pairs of orbits
created at period-doubling bifurcations of touch-and-go type are given by two
different linear combinations of algebraic Lam\'e functions with period 8K.Comment: LaTeX2e, 22 pages, 14 figures. Version 3: final form of paper,
accepted by J. Phys. A. Changes in Table 2; new reference [25]; name of
bifurcations "touch-and-go" replaced by "island-chain
On a q-difference Painlev\'e III equation: I. Derivation, symmetry and Riccati type solutions
A q-difference analogue of the Painlev\'e III equation is considered. Its
derivations, affine Weyl group symmetry, and two kinds of special function type
solutions are discussed.Comment: arxiv version is already officia
New Two-Dimensional Integrable Quantum Models from SUSY Intertwining
Supersymmetrical intertwining relations of second order in the derivatives
are investigated for the case of supercharges with deformed hyperbolic metric
. Several classes of particular solutions of these
relations are found. The corresponding Hamiltonians do not allow the
conventional separation of variables, but they commute with symmetry operators
of fourth order in momenta. For some of these models the specific SUSY
procedure of separation of variables is applied.Comment: 18 page
Intertwining relations of non-stationary Schr\"odinger operators
General first- and higher-order intertwining relations between non-stationary
one-dimensional Schr\"odinger operators are introduced. For the first-order
case it is shown that the intertwining relations imply some hidden symmetry
which in turn results in a -separation of variables. The Fokker-Planck and
diffusion equation are briefly considered. Second-order intertwining operators
are also discussed within a general approach. However, due to its complicated
structure only particular solutions are given in some detail.Comment: 18 pages, LaTeX20
Moment inversion problem for piecewise D-finite functions
We consider the problem of exact reconstruction of univariate functions with
jump discontinuities at unknown positions from their moments. These functions
are assumed to satisfy an a priori unknown linear homogeneous differential
equation with polynomial coefficients on each continuity interval. Therefore,
they may be specified by a finite amount of information. This reconstruction
problem has practical importance in Signal Processing and other applications.
It is somewhat of a ``folklore'' that the sequence of the moments of such
``piecewise D-finite''functions satisfies a linear recurrence relation of
bounded order and degree. We derive this recurrence relation explicitly. It
turns out that the coefficients of the differential operator which annihilates
every piece of the function, as well as the locations of the discontinuities,
appear in this recurrence in a precisely controlled manner. This leads to the
formulation of a generic algorithm for reconstructing a piecewise D-finite
function from its moments. We investigate the conditions for solvability of the
resulting linear systems in the general case, as well as analyze a few
particular examples. We provide results of numerical simulations for several
types of signals, which test the sensitivity of the proposed algorithm to
noise
A nonlocal connection between certain linear and nonlinear ordinary differential equations/oscillators
We explore a nonlocal connection between certain linear and nonlinear
ordinary differential equations (ODEs), representing physically important
oscillator systems, and identify a class of integrable nonlinear ODEs of any
order. We also devise a method to derive explicit general solutions of the
nonlinear ODEs. Interestingly, many well known integrable models can be
accommodated into our scheme and our procedure thereby provides further
understanding of these models.Comment: 12 pages. J. Phys. A: Math. Gen. 39 (2006) in pres
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