Volume 33, Issue 11, November 1992
Index of content:

The pattern calculus for tensor operators in quantum groups
View Description Hide DescriptionAn explicit algebraic evaluation is given for all q‐tensor operators (with unit norm) belonging to the quantum groupU _{ q } ( u(n)) and having extremal operator shift patterns acting on arbitrary U _{ q } ( u(n)) irreps. These rather complicated results are shown to be easily comprehensible in terms of a diagrammatic calculus of patterns. A more conceptual derivation of these results is discussed using fundamental properties of the q−6j coefficients.

On the q‐symplecton realization of the quantum group SU_{ q }(2)
View Description Hide DescriptionA q‐symplecton realization of the quantum groupSU_{ q }(2) is constructed, giving explicitly the product law for the Weyl‐ordered characteristic polynomials. A new form of the q‐Racah coefficients is obtained, which is especially adapted to proving a new symmetry for these functions. A surprising result is that unlike the SU(2) case there are two distinct invariant q‐triangle functions in the q‐symplecton extension.

The Fermi and Bose congruences for free semigroups on two generators
View Description Hide DescriptionUsing a free semigroup and a free monoid generated by two symbols, two congruences are introduced. These congruences provide natural equivalence classifications of all finite composite creation and annihilation operators for fermions and bosons such that equivalent operators act identically (to within a nonnegative integer multiple) upon the associated basis states. The resulting quotient structures are shown to be isomorphic to two elementary inverse semigroups and their algebras are used to construct generalized commutation forms. These provide for the easy evaluation of any fermion anticommutator and any reduced word boson commutator.

Self‐adjointness of the operators in Wigner’s commutation relations
View Description Hide DescriptionBoth self‐adjointness and spectra of the momentum operator p and the Hamiltonian H are studied: p=−i{d/dx−(c/x)R}; H=2^{−1}{−(d/dx)^{2}+(c/x ^{2})(c−R)+x ^{2}}, for a one‐dimensional harmonic oscillator governed by Wigner’s commutation relations. Here, c is a real parameter and Rψ(x)=ψ(−x). First the essential self‐adjointness of p and H are shown to obtain that the spectrum of H consists of eigenvalues only. Moreover, we clarify the meaning in L ^{2}(R) of the classical form H=2^{−1}(p ^{2}+x ^{2}). Next it is shown that every real number is in the continuous spectrum of p, by introducing a kind of generalized Fourier transformB satisfying p=B ^{−1} xB for the multiplication x. In the case of c=0, B turns out to be the ordinary Fourier transform and p becomes the well‐known operator. By the methods stated above, all the results of the recent papers, where some restrictions on the parameter c were imposed, are improved.

On a zero curvature problem related to the ZS–AKNS operator
View Description Hide DescriptionAll the ZS–AKNS operators, L = i[^{∂ x −g(x)} _{ r(x)−∂ x }], which admit two linearly independent families of eigenfunctions, satisfying a differential equation in the spectral parameter λ of the form a(λ)∂_{λ} V+B(λ)V=Θ(x)V, are characterized. Here, a(λ)∈C, B, and Θ take values in the space of 2×2 complex matrices.

New reductions of the Kadomtsev–Petviashvili and two‐dimensional Toda lattice hierarchies via symmetry constraints
View Description Hide DescriptionNew types of reductions of the Kadomtsev–Petviashvili (KP) hierarchy and the two‐dimensional Toda lattice (2DTL) hierarchy are considered on the basis of Sato’s approach. Within this approach these hierarchies are represented by infinite sets of equations for potentials u _{1},u _{2},u _{3},..., of pseudodifferential operators and their eigenfunctions ψ and adjoint eigenfunctions ψ*. The KP and the 2DTL hierarchies are studied under constraints of the following type: ∑_{ n=1} ^{ N } α_{ nS } _{ n }(u _{1},u _{2},u _{3},...)=Ω_{ x }, where S _{ n } are symmetries for these hierarchies, α_{ n } are arbitrary constants, and Ω is an arbitrary linear functional of the quantity ψ(λ)ψ*(μ). It is shown that for the KP hierarchy these constraints give rise to hierarchies of 1+1‐dimensional commuting flows for the variables u _{2},u _{3},...,u _{ N },ψ,ψ*. Many known systems and several new ones are among them. Symmetry reductions for the 2DTL hierarchy give rise both to finite‐dimensional dynamical systems and 1+1‐dimensional discrete systems. Some few results for the modified KP hierarchy are also presented.

On symmetries of radially symmetric nonlinear diffusion equations
View Description Hide DescriptionIn this paper symmetries are sought for radially symmetric nonlinear diffusion equations of the type u _{ t }=r ^{1−N }[r ^{ N−1} f(u)u _{ r }]_{ r }. The nonlinear forms of this equation which admit Lie classical symmetries are completely classified and are shown to exist when f=u ^{ n }, e ^{ u }. Also, a trivial symmetry exists for arbitrary f. Lie–Bäcklund symmetries are also considered when f=u ^{ n } and shown to exist when n=0 (linear case) and when n=−2 and N=1.

Quasiperiodic solutions for matrix nonlinear Schrödinger equations
View Description Hide DescriptionThe Adler–Kostant–Symes theorem yields isospectral Hamiltonian flows on the dual ■erm>$$̃g^{+}* of a Lie subalgebra ■erm>$$̃g^{+} of a loop algebra ■erm>$$̃g. A general approach relating the method of integration of Krichever, Novikov, and Dubrovin to such flows is used to obtain finite‐gap solutions of matrix nonlinear Schrödinger equations in terms of quotients of θ functions.

The universal propagator for affine [or SU(1,1)] coherent states
View Description Hide DescriptionIt is shown that it is possible to introduce a universal propagator for a general Hamiltonian appropriate to a single affine degree of freedom. This universal propagator is a single function, independent of any particular choice of fiducial vector, which, nonetheless, propagates the coherent stateHilbert space representatives correctly. Furthermore, the universal propagator for several examples is explicitly constructed.

R‐matrix approach to quantum superalgebras su_{ q }(m‖n)
View Description Hide DescriptionQuantum superalgebras su_{ q }(m‖n) are studied in the framework of R‐matrix formalism. Explicit parametrization of L ^{(+)} and L ^{(−)} matrices in terms of su_{ q }(m‖n) generators are presented. We also show that quantum deformation of nonsimple superalgebra su(n‖n) requires its extension to u(n‖n).

Conditions for a symmetric connection to be a metric connection
View Description Hide DescriptionA recent result−in ‘general circumstances’ for a four‐dimensional space–time−giving algebraic conditions on a curvature tensor (of a symmetric connection) so that the connection be metric, is shown to be a special case of a more general result; both of these results are shown formally to be of a generic nature. In this new result the conditions are imposed on a tensor of a more general character than the curvature tensor. In addition it is shown that once the symmetric connection is known to be metric, the metric is uniquely defined (up to a constant conformal factor). For those special curvature tensors which are excluded from the original result, supplementary conditions are suggested, which, alongside the original conditions are sufficient to ensure that most of these excluded curvature tensors are also Riemann tensors.

Natural bundles. I. A minimal resolution of superspace
View Description Hide DescriptionIn this, the first of a sequence of papers applying infinite dimensional natural bundle techniques to problems in general relativity and other field theories, we construct a minimal resolution of the singularities in the space of geometries or superspace of a closed manifoldM. The basic properties of two natural bundles associated with M are also discussed. Subsequent articles will develop the theory in the context of spin and the diffeomorphism group, the Yang–Mills moduli space, and nonlinear sigma models.

Reduced description of electric multipole potential in Cartesian coordinates
View Description Hide DescriptionThe electrostatic potential due to a multipole moment of order l is expressed in terms of 2l+1 independent Cartesian multipole components. The multipole expansion of the electrostatic interaction energy between two charge distributions is reduced correspondingly to the minimum number of Cartesian components.

Quasiperiodic systems without Cantor‐set‐like energy bands
View Description Hide DescriptionA class of quasiperiodic systems is proposed that does not show the Cantor‐set‐like energy bands. The general aspects of such systems are investigated.

Approximate analytic solutions to the Skyrme equation
View Description Hide DescriptionThe asymptotic properties and the stability of the solutions of the Skyrme nonlinear differential equation for the profile function F are discussed. Using rational approximants for σ=cos F some different ways of finding approximate analytic solutions to the equation are discussed. For boundary conditions appropriate to soliton number 1 we construct a sequence of approximate analytic solutions that rapidly approaches the true solution according to several quality tests.

Bi‐Hamiltonian systems and Maslov indices
View Description Hide DescriptionIt is shown that any Lagrangian distribution on a 2n‐dimensional symplectic manifold Γ is equivalent to a principal O(n) bundle obtained as a reduction of the frame bundle. Bisymplectic manifolds with Nijenhuis recursion operators are studied and it is shown that the set of all bi‐Lagrangian distributions is a trivial bundle the structure group being the homogeneous space U(1)^{ n }/O(1)^{ n }. Various formulas for Maslov indices of closed curves are given including one using only data from the recursion operator.

On the solution of homogeneous generalized wave equation
View Description Hide DescriptionWe present the solutions for the homogeneous generalized wave equation when we have a local problem.

Line soliton interactions of a nonisospectral and variable‐coefficient Kadomtsev–Petviashvili equation
View Description Hide DescriptionAn extension of the dressing method is developed to solve a nonisospectral and variable coefficient generalization of the Kadomtsev–Petviashvili equation. Nonstandard dynamics of the one‐soliton solutions are presented. Decomposition of the two‐solitons and their interactions are studied in detail.

Constraints of the Kadomtsev–Petviashvili hierarchy
View Description Hide DescriptionFor the Kadomtsev–Petviashvili (KP) hierarchy constructed in terms of the famous Sato theory, a ‘‘k constraint’’ is proposed that leads the hierarchy to the nonlinear system involving a finite number of dynamical coordinates. The eigenvalue problem of the k‐constrained system is naturally obtained from the linear system of the KP hierarchy, which takes the form of kth‐order polynomial coupled with a first‐order one, thus we are able to derive the correspondent Lax pair, recursion operator, bi‐Hamiltonian structures, and conserved quantities. The constraints for the BKP hierarchy are also sketched.

Generalized intermediate long‐wave hierarchy in zero‐curvature representation with noncommutative spectral parameter
View Description Hide DescriptionThe simplest generalization of the intermediate long‐wave hierarchy (ILW) is considered to show how to extend the Zakharov–Shabat dressing method to nonlocal, i.e., integro‐partial differential, equations. The purpose is to give a procedure of constructing the zero‐curvature representation of this class of equations. This result obtains by combining the Drinfeld–Sokolov formalism together with the introduction of an operator‐valued spectral parameter, namely, a spectral parameter that does not commute with the space variable x. This extension provides a connection between the ILW_{ k } hierarchy and the Saveliev–Vershik continuum graded Lie algebras. In the case of ILW_{2} the Fairlie–Zachos sinh‐algebra was found.