Free-field realization of theWKP(N) algebra

TheW _KP ^(N) algebra has been identified with the second Hamiltonian structure in theNth Hamiltonian pair of the KP hierarchy. In this Letter, by constructing the Miura map that decomposes the second Hamiltonian structure in theNth pair of the KP hierarchy, we show thatW _KP ^(N) can also be decomposed toN independent copies ofW _KP ⁽¹⁾ algebras, therefore its free-field realization can be worked out by constructing free fields for each copy ofW _KP ⁽¹⁾ . In this way, the free fields may consist ofN + 2n number of bosons, among them, 2n are in pairs, wheren is an arbitrary integer between 1 andN. We also express the currents ofW _KP ^(N) in terms of the currents ofN —n copies of U(1) andn copies of SL(2,R) _k algebras with levelk = 1. By reductions, we give similar results forW ^(N) andW ₃ ⁽²⁾ algebra.     

On generalized Lax equation of the Lax triple of KP Hierarchy

In terms of the operator Nambu 3-bracket and the Lax pair (L, B_n) of the KP hierarchy, we propose the generalized Lax equation with respect to the Lax triple (L, B_n, B_m). The intriguing results are that we derive the KP equation and another integrable equation in the KP hierarchy from the generalized Lax equation with the different Lax triples (L, B_n, B_m). Furthermore we derive some no integrable evolution equations and present their single soliton solutions.     

The algebra of differential operators on the circle and W KP (q)

Radul has recently introduced a map from the Lie algebra of differential operators on the circle of W _n . In this Letter, we extend this map to W _KP ^(q) , a recently introduced one-parameter deformation of W_KP - the second Hamiltonian structure of the KP hierarchy. We use this to give a short proof that W is the algebra of additional symmetries of the KP equation.     

The integrable hierarchy constructed from a pair of KdV-type hierarchies and its associatedW algebra

For any two arbitrary positive integers n and m, using them ^th KdV hierarchy and the (n+m)^th KdV hierarchy as building blocks, we are able to construct another integrable hierarchy (referred to as the (n, m)^th KdV hierarchy). TheW-algebra associated to the second Hamiltonian structure of the (n, m)^th KdV hierarchy (calledW(n, m) algebra) is isomorphic via a Miura map to the direct sum of aW _m -algebra, aW _n+m -algebra and an additionalU(1) current algebra. In turn, from the latter, we can always construct a representation of aW -algebra.     

Remarks on Hamiltonian structures of the KP hierarchy and W KP( q ) algebra

It is known that second Hamiltonian structures of the KP hierarchy are parameterized by a continuous complex parameter q and correspond to the W-infinite algebra of W _infKP ^sup(q) . In this Letter, by constructing a Miura map, we first show a generalized decomposition theorem to the second Hamiltonian structures and then establish a relation between those structures which corresponds to values (q+1) and q of the parameter, respectively. This discussion also gives a better understanding to the structures of W _infKP ^sup(q) , its reduced algebras, and their free fields realizations.     

P-adic theta functions and solutions of the KP hierarchy

Based on Schottky uniformization theory of Riemann surfaces, we construct a universal power series for (Riemann) theta function solutions of the KP hierarchy. Specializing this power series to the coordinates associated with Schottky groups overp-adic fields, we show that thep-adic theta functions of Mumford curves give solutions of the KP hierarchy.     

Role of localized excitons in the stimulated emission in ultra-thin CdSe/ZnSe/ZnSSe single quantum-well structures

Photo-pumped lasing properties have been investigated in a CdSe/ZnSe/ZnSSE single quantum wells (SQWs) with the well-layer thickness (L_W) of 1, 2 and 3 monolayer (ML). At 20 K, the laser threshold for the SQW withL_W = 1 ML was the lowest in spite of the smallest active layer thickness. The carrier (exciton) sheet density at the threshold (n)_thwas estimated to be as low as 7 × 10¹⁰cm⁻², which is well below Mott's screening density. Time-resolved photoluminescence has revealed that the localized biexciton band, whose peak energy agrees with the lasing peak, appeared on the low-energy side of the main PL peak at this level of carrier concentrations. Theoretical calculation has also shown that the localized biexciton recombination has to be taken into account for the lasing process. On the contrary, then_thvalues of the SQWs with 2 and 3 ML are 1 order of magnitude larger than that of the SQW with 1 ML. This may be due to the smaller oscillator strength of both localized excitons and localized biexcitons because of the larger inhomogeneous broadening, resulting in an increased carrier density for achieving optical gain sufficient to overcome the reflection losses.     

Extended Discrete KP Hierarchy and its Reductions from a Geometric Viewpoint

We interpret the recently suggested extended discrete KP (Toda lattice) hierarchy from a geometrical point of view. We show that the latter corresponds to the union of invariant submanifolds S ₀ ⁿ of the system which is a chain of infinitely many copies of Darboux–KP hierarchy, while the intersections yields a number of reduction s to l-field lattices.     

Analogs of Gallagher-Moszkowski bands in IBFFM — A relation to spectrum-generating supersymmetry?

The quantal analogs of the Gallagher-Moszkowski bands in the Interacting boson-fermion-fermion model (IBFFM) withSU (3) boson core are associated with the exact quantum numberKj _p+j_n and approximate quantum numberK-¦j _p-j_n¦ of IBFFM. For characteristic ratiosГ/δ it is shown that theK _c=0 band in even-even IBM system, theK=j _p andK=j _n bands in the corresponding odd-even (IBFM) systems and theK=j _p+j band in the corresponding odd-odd IBFFM system exhibit the supermultiplet pattern.     

Group theory and kink stability equations

Starting from Schrödinger equations withSU(2) group-theoretic potentials, we consider a general family of kinks labeled by two (half-)integers (l, n) with ¦n¦l. A particular choice ofn=0,l=L (L positive integer) leads to a generalL-family, whereL=1 corresponds to sine-Gordon theory, whileL=2 represents the ( ⁴)₁₊₁ model. The ( ⁶)₁₊₁ model can also be recovered withl=3/2,n=–1/2, a particular case of theories labeled byl andn such thatl-n=2 which possess simple kink solutions. We also discuss one-loop order corrections to the kink masses in supersymmetric versions of theL-family. As a byproduct, we obtain the SUSY renormalization of the so-called parameter in sine-Gordon theory.     

Hirota polynomials for the KP and BKP hierarchies

Using symmetric function techniques, we derive closed-form expressions for the Hirota polynomials for thepth modified KP and BKP hierarchies in terms of Schur and SchurQ-polynomials, respectively. The Hirota polynomials for the BKP hierarchy can also be expressed as Pfaffians while those for thepth modified KP hierarchies can, under certain conditions, be expressed as determinants.     

Mössbauer study of the hyperfine and magnetic properties of organo-di-iron electron reservoirs containing a polyaromatic bridging ligand

Organo di-iron electron reservoirs Fe(CP^*)₂(Ar) ⁿ⁺ withn=2, 1, 0, where Cp^* is C₅(CH₃)₅ and where Ar are the following bridges: biphenyl, dihydrophenanthrene, triphenylene, have been studied by Mössbauer spectroscopy in the solid state. Complexes withn=2, with 36e^? in the coordination spheres of the metals, exhibit the usual diamagnetic behaviour of 18e^?, Fe^II mono-iron systems. Complexes withn=1, 37e^?, are delocalized mixed valence (Fe^IIFe^I) with a spin 1/2; the magnetic hyperfine interaction, measured under an external field, shows equal delocalization of the 37^th e^? on the two iron centers and the two bridging carbon atoms of the biphenylene. Complexes withn=0, formally with 38e^?, have a practically temperature-independent quadrupole splitting, and isomer shift values which constrast with the expected behaviour of independent Fe^I, 19e^? centers. This indicates that the 37^th and 38^th electrons are mostly located on the polyaromatic bridge. Spectra obtained in an external field show a negligible magnetic hyperfine interaction and support this conclusion. In the case of biphenyl and dihydrophenanthrene bridges, this electron localization can be related to a strong intramolecular chemical coupling, evidenced by other spectroscopic and X-ray data [1].     

Solving the constrained modified KP hierarchy by gauge transformations

In this paper, we mainly investigate two kinds of gauge transformations for the constrained modified KP hierarchy in Kupershmidt-Kiso version. The corresponding gauge transformations are required to keep not only the Lax equation but also the Lax operator. For this, by selecting the special generating eigenfunction and adjoint eigenfunction, the elementary gauge transformation operators of modified KP hierarchy T_D(Φ) = (Φ^?1)^?_x¹? Φ^?1 and T_I (Ψ) = Ψ^?1? ^?1Ψ_x, become the ones in the constrained case. Finally, the corresponding successive applications of T_D and T_I on the eigenfunction Φ and the adjoint eigenfunction Ψ are discussed.     

Bilinear Identities and Hirota’s Bilinear Forms for the (γn, σk)-KP Hierarchy

In this paper, we discuss how to construct the bilinear identities for the wave functions of the (γ_n, σ_k)-KP hierarchy and its Hirota’s bilinear forms. First, based on the corresponding squared eigenfunction symmetry of the KP hierarchy, we prove that the wave functions of the (γ_n, σ_k)-KP hierarchy are equal to the bilinear identities given in Sec.3 by introducing N auxiliary parameters z_i, i = 1, 2,?…?, N. Next, we derived the bilinear equations for the tau-function of the (γ_n, σ_k)-KP hierarchy. Then, we obtain the bilinear equations for the taufunction of the mixed type of KP equation with self-consistent sources (KPESCS), which includes both the first and the second type of KPESCS as special cases by setting n = 2 and k = 3. Finally, using the relation between the Hirota bilinear derivatives and the usual partial derivatives, we show the procedure of translating the Hirota’s bilinear equations into the mixed type of KPESCS.     

Classification of Simple Linearly Compact n-Lie Superalgebras

We classify simple linearly compact n-Lie superalgebras with n > 2 over a field ${\mathbb{F}}We classify simple linearly compact n-Lie superalgebras with n > 2 over a field \mathbbF{\mathbb{F}} of characteristic 0. The classification is based on a bijective correspondence between non-abelian n-Lie superalgebras and transitive \mathbbZ{\mathbb{Z}}-graded Lie superalgebras of the form L=?_j=-1^n-1 L_j{L=\oplus_{j=-1}^{n-1} L_j}, where dim L _n−1 = 1, L ₋₁ and L _n−1 generate L, and [L _j , L _n−j−1] = 0 for all j, thereby reducing it to the known classification of simple linearly compact Lie superalgebras and their \mathbbZ{\mathbb{Z}}-gradings. The list consists of four examples, one of them being the n + 1-dimensional vector product n-Lie algebra, and the remaining three infinite-dimensional n-Lie algebras.     

Generalized Drinfel'd-Sokolov hierarchies

In this paper we examine the bi-Hamiltonian structure of the generalized KdV-hierarchies. We verify that both Hamiltonian structures take the form of Kirillov brackets on the Kac-Moody algebra, and that they define a coordinated system. Classical extended conformal algebras are obtained from the second Poisson bracket. In particular, we construct theW _n ^(l) algebras, first discussed for the casen=3 andl=2 by Polyakov and Bershadsky.     

Critical exponents of Manhattan Hamiltonian walks in two dimensions,from Potts andO(n) models

We consider a set of Hamiltonian circuits filling a Manhattan lattice, i.e., a square lattice with alternating traffic regulation. We show that the generating function (with fugacityz) of this set is identical to the critical partition function of aq-state Potts model on an unoriented square lattice withq ^1/2 =z. The set of critical exponents governing correlations of Hamiltonian circuits is derived using a Coulomb gas technique. These exponents are also found to be those of an O(n) vector model in the low-temperature phase withn =q ^1/2 =z. The critical exponents in the limitz = 0 are then those of spanning trees (q= 0) and of dense polymers (n=0,T < T_c), corresponding to a conformal theory with central chargeC = –2. This shows that the Manhattan orientation and the Hamiltonian constraint of filling all the lattice are irrelevant for the infrared critical properties of Hamiltonian walks.     

The coherent potential approximation is a realizable effective medium scheme

The effective electrical conductivity of an aggregate, composed of grains of various conductivities, is frequently estimated by the coherent potential approximation, which embodies a local effective medium concept. It is proved rigorously that this approximation is exact for a wide class of hierarchical model composites made of spherical grains: the starting material 0 in the hierarchy is chosen arbitrarily, otherwise, materialj=1, 2, ... consists of equisized spheres, sayj-spheres, of arbitrary conductivities embedded in materialj — 1. The spatial distribution of thej-spheres must satisfy a mild homogeneity condition and their radiusr _j must, asymptotically, increase faster than exponentially withj. Furthermore, the minimum spacing, 2s _j , between thej-spheres is such that the ratios _j /r _j diverges. On the basis of these and some further ancillary conditions it is established that the coherent potential approximation becomes asymptotically exact for the effective conductivity of materialj. The results extend to other effective parameters of the composites, including the thermal conductivity, dielectric constant and magnetic permeability. In addition, the model composites and the proof of realizability may be generalized to allow non-spherical grains.     

Construction of positive exact (2+1)-dimensional shock wave solutions for two discrete Boltzmann models

It is proved that (2+1)-dimensional (spacex, y; timet) positive exact shock wave solutions of two discrete Boltzmann models exist. For each densityN _i, these solutions are linear combinations of three similarity shock waves,N _i =n _0i + _j n _ji /[1+d _j exp( _j y+y _j x+ _j t)],j=1,2,3. Two models with four independent densities are investigated: the square discrete-velocity Boltzmann model and the model with eight velocities oriented toward the eight corners of a cube.The positivity problem for the densities is nontrivial. Two classes of solutions are considered for which the two first similarity shock wave components depend on only one spatial dimension, _j=const· _j ,j=1,2. For the positivity, if ₁₂>0, it is sufficient to prove that the 16 asymptotic shock limitsn _0i ,n _0i +n _3i , _j=0 ² n _ji , _j=0 ³ n _ji are positive. The density solutions are built up with five arbitrary parameters and we prove that there exist subdomains of the arbitrary parameter space in which the 16 shock limits are positive. We study numerically two explicit shock wave solutions. We are interested in the movement of the shock front when the time is growing and in the possible appearance of bumps. In the space, at intermediate times, these bumps represent populations of particles which are larger than at initial time or at equilibrium time.