2,501 research outputs found
N=1,2 Super-NLS Hierarchies as Super-KP Coset Reductions
We define consistent finite-superfields reductions of the super-KP
hierarchies via the coset approach we already developped for reducing the
bosonic KP-hierarchy (generating e.g. the NLS hierarchy from the
coset). We work in a manifestly supersymmetric framework
and illustrate our method by treating explicitly the super-NLS
hierarchies. W.r.t. the bosonic case the ordinary covariant derivative is now
replaced by a spinorial one containing a spin
superfield. Each coset reduction is associated to a rational super-\cw
algebra encoding a non-linear super-\cw_\infty algebra structure. In the
case two conjugate sets of superLax operators, equations of motion and
infinite hamiltonians in involution are derived. Modified hierarchies are
obtained from the original ones via free-fields mappings (just as a m-NLS
equation arises by representing the algebra through the
classical Wakimoto free-fields).Comment: 27 pages, LaTex, Preprint ENSLAPP-L-467/9
Coisotropic deformations of associative algebras and dispersionless integrable hierarchies
The paper is an inquiry of the algebraic foundations of the theory of
dispersionless integrable hierarchies, like the dispersionless KP and modified
KP hierarchies and the universal Whitham's hierarchy of genus zero. It stands
out for the idea of interpreting these hierarchies as equations of coisotropic
deformations for the structure constants of certain associative algebras. It
discusses the link between the structure constants and the Hirota's tau
function, and shows that the dispersionless Hirota's bilinear equations are,
within this approach, a way of writing the associativity conditions for the
structure constants in terms of the tau function. It also suggests a simple
interpretation of the algebro-geometric construction of the universal Whitham's
equations of genus zero due to Krichever.Comment: minor misprints correcte
Supersymmetric quantum mechanics and Painleve equations
In these lecture notes we shall study first the supersymmetric quantum
mechanics (SUSY QM), specially when applied to the harmonic and radial
oscillators. In addition, we will define the polynomial Heisenberg algebras
(PHA), and we will study the general systems ruled by them: for zero and first
order we obtain the harmonic and radial oscillators, respectively; for second
and third order PHA the potential is determined by solutions to Painleve IV
(PIV) and Painleve V (PV) equations. Taking advantage of this connection, later
on we will find solutions to PIV and PV equations expressed in terms of
confluent hypergeometric functions. Furthermore, we will classify them into
several solution hierarchies, according to the specific special functions they
are connected with.Comment: 38 pages, 20 figures. Lecture presented at the XLIII Latin American
School of Physics: ELAF 2013 in Mexico Cit
Conformal Mappings and Dispersionless Toda hierarchy
Let be the space consists of pairs , where is a
univalent function on the unit disc with , is a univalent function
on the exterior of the unit disc with and
. In this article, we define the time variables , on which are holomorphic with respect to the natural
complex structure on and can serve as local complex coordinates
for . We show that the evolutions of the pair with
respect to these time coordinates are governed by the dispersionless Toda
hierarchy flows. An explicit tau function is constructed for the dispersionless
Toda hierarchy. By restricting to the subspace consists
of pairs where , we obtain the integrable hierarchy
of conformal mappings considered by Wiegmann and Zabrodin \cite{WZ}. Since
every homeomorphism of the unit circle corresponds uniquely to
an element of under the conformal welding
, the space can be naturally
identified as a subspace of characterized by . We
show that we can naturally define complexified vector fields \pa_n, n\in \Z
on so that the evolutions of on
with respect to \pa_n satisfy the dispersionless Toda
hierarchy. Finally, we show that there is a similar integrable structure for
the Riemann mappings . Moreover, in the latter case, the time
variables are Fourier coefficients of and .Comment: 23 pages. This is to replace the previous preprint arXiv:0808.072
Stationary problems for equation of the KdV type and dynamical -matrices.
We study a quite general family of dynamical -matrices for an auxiliary
loop algebra related to restricted flows for equations of
the KdV type. This underlying -matrix structure allows to reconstruct Lax
representations and to find variables of separation for a wide set of the
integrable natural Hamiltonian systems. As an example, we discuss the
Henon-Heiles system and a quartic system of two degrees of freedom in detail.Comment: 25pp, LaTe
Fusion hierarchies, T-systems and Y-systems of logarithmic minimal models
A Temperley-Lieb (TL) loop model is a Yang-Baxter integrable lattice model
with nonlocal degrees of freedom. On a strip of width N, the evolution operator
is the double-row transfer tangle D(u), an element of the TL algebra TL_N(beta)
with loop fugacity beta=2cos(lambda). Similarly on a cylinder, the single-row
transfer tangle T(u) is an element of the enlarged periodic TL algebra. The
logarithmic minimal models LM(p,p') comprise a subfamily of the TL loop models
for which the crossing parameter lambda=(p'-p)pi/p' is parameterised by coprime
integers 0<p<p'. For these special values, additional symmetries allow for
particular degeneracies in the spectra that account for the logarithmic nature
of these theories. For critical dense polymers LM(1,2), D(u) and T(u) are known
to satisfy inversion identities that allow us to obtain exact eigenvalues in
any representation and for all system sizes N. The generalisation for p'>2
takes the form of functional relations for D(u) and T(u) of polynomial degree
p'. These derive from fusion hierarchies of commuting transfer tangles
D^{m,n}(u) and T^{m,n}(u) where D(u)=D^{1,1}(u) and T(u)=T^{1,1}(u). The fused
transfer tangles are constructed from (m,n)-fused face operators involving
Wenzl-Jones projectors P_k on k=m or k=n nodes. Some projectors P_k are
singular for k>p'-1, but we argue that D^{m,n}(u) and T^{m,n}(u) are well
defined for all m,n. For generic lambda, we derive the fusion hierarchies and
the associated T- and Y-systems. For the logarithmic theories, the closure of
the fusion hierarchies at n=p' translates into functional relations of
polynomial degree p' for D^{m,1}(u) and T^{m,1}(u). We also derive the closure
of the Y-systems for the logarithmic theories. The T- and Y-systems are the key
to exact integrability and we observe that the underlying structure of these
functional equations relate to Dynkin diagrams of affine Lie algebras.Comment: 77 page
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