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Jan 1991

Volume 32, Issue 1, pp. 1-317

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Graph theory and a special class of symplectic transformations: The generalized Jacobi variables

Chjan C. Lim

J. Math. Phys. 32, 1 (1991); http://dx.doi.org/10.1063/1.529119 (7 pages) | Cited 2 times

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The well‐known Jacobi variables in celestial mechanics are generalized to other Hamiltonian systems which include vortex dynamics. A combinatorial algorithm for constructing the generalized Jacobi variables is given; for any binary tree T(N) with N leave, there is a 2N×2N real symplectic matrix MT (T(N),Γ) which completely defines a linear canonical transformation to these relative variables. This algorithm yields a direct proof of the symplectic property for all the generalized Jacobi variables. An application to vortex dynamics is outlined here.
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02.10.Ud Linear algebra
02.10.Xm Multilinear algebra
45.05.+x General theory of classical mechanics of discrete systems
02.60.Gf Algorithms for functional approximation

Integrable heterogeneous nonlinear Schrödinger equations with dielectric loss: Lie–Bäcklund symmetries

P. Broadbridge and S. E. Godfrey

J. Math. Phys. 32, 8 (1991); http://dx.doi.org/10.1063/1.529099 (11 pages) | Cited 3 times

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The general cubic nonlinear Schrödinger equation, with heterogeneity and dielectric loss, is iψt=− (1)/(2) ψxx+V(x,t)ψ+iU(x,t)ψ+R (x,t)‖ψ‖2ψ. All such equations are found that possess a third‐order Lie–Bäcklund symmetry. Each of these equations are then transformed to the standard completely integrable cubic Schrödinger equation (U=V=0, R=±1). In addition, some useful structure results are established for the time‐dependent symmetries of the linear 2+1‐dimensional Schrödinger equation with arbitrary real potential.
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02.20.-a Group theory
03.65.Fd Algebraic methods

Representations of the SL(2,C) Kac–Moody current algebra in the Fock space

Z. Haba

J. Math. Phys. 32, 19 (1991); http://dx.doi.org/10.1063/1.529118 (5 pages) | Cited 1 time

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A representation of the SL(2,C) current algebra in terms of Hermitian currents constructed from free fields is obtained. The representation is not of the highest weight type. The restriction k≥4 on the Kac–Moody central charge k is obtained. If k=4, then the Sugawara construction defines the energy‐momentum tensor equal to the energy‐momentum tensor of the free‐field theory.
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02.20.-a Group theory
03.70.+k Theory of quantized fields
11.40.-q Currents and their properties

Even sectors of Lie superalgebras as locally convex Lie algebras

Vladimir Pestov

J. Math. Phys. 32, 24 (1991); http://dx.doi.org/10.1063/1.529126 (9 pages) | Cited 4 times

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Let g be a finite‐dimensional Lie superalgebra over a graded‐commutative local augmented Arens–Michael ground algebra Λ. A Lie group is associated with a number of pleasant properties to the even sector @Fg0 viewed as an ordinary complete locally convex Lie algebra (which is in general infinite dimensional).
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02.20.-a Group theory
02.40.-k Geometry, differential geometry, and topology
04.65.+e Supergravity

On the classification of Lie subalgebras and Lie subgroups

D. R. Grigore and O. T. Popp

J. Math. Phys. 32, 33 (1991); http://dx.doi.org/10.1063/1.529091 (7 pages) | Cited 2 times

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Starting from a paper of Patera et al. [J. Math. Phys. 16, 1597 (1975)], the classification of Lie subalgebras is analyzed, up to conjugation, for a Lie algebra having a solvable ideal. Here, the hypothesis of semidirect product structure is not made; this leads to some natural affine structure. Next, a complete classification is made of closed Lie subgroups of a given Lie group, up to conjugation (supposing the corresponding classification for Lie subalgebras is known). This is illustrated in the case of the group SE(3).
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02.20.Sv Lie algebras of Lie groups

On the structure and properties of the singularity manifold equations of the KP hierarchy

Boris Konopelchenko and Walter Strampp

J. Math. Phys. 32, 40 (1991); http://dx.doi.org/10.1063/1.529505 (10 pages) | Cited 14 times

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This paper is concerned with the Painlevé expansion and the singularity manifold (SM) equation of the Kadomtsev–Petviashvili (KP) equation. Several aspects of the interrelation between the SM equation and the KP auxiliary linear system are studied. It is shown that the simultaneous Painlevé expansion for the KP potential u and the KP eigenfunction ψ can be treated as a Bäcklund‐gauge transformation. Two methods for the derivation of the SM equation based on this treatment are proposed and their equivalence is proved. The interrelation between the SM equation and the vertical hierarchy of the KP eigenfunction equations is discussed. The explanation of the coincidence of the KP eigenfunction equation of the second level and the KP SM equation is given. Compact forms of the hierarchy of SM equations of the KP hierarchy are presented. The connection between the KP singularity manifold function φ and the KP eigenfunctions ψ and the adjoint KP eigenfunctions ψ∗ is derived. The bilinear‐bilocal description of the hierarchy the KP SM equations is given within the framework of Sato’s τ‐function theory.
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02.30.-f Function theory, analysis
41.20.Jb Electromagnetic wave propagation; radiowave propagation
03.65.-w Quantum mechanics

Three‐body plane wave at zero angular momentum and some addition theorems

A. A. Kvitsinsky and S. P. Merkuriev

J. Math. Phys. 32, 50 (1991); http://dx.doi.org/10.1063/1.529092 (7 pages) | Cited 2 times

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A quantum three‐body system with zero angular momentum is considered. The plane wave F related to this problem is studied. It is proved to be a function of two variables that have a meaning of the eikonals on the internal space. A number of explicit formulas for F and its asymptotics are derived. New addition theorems for the scalar hyperspherical harmonics as well as for some special functions are obtained.
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02.30.-f Function theory, analysis
03.65.Nk Scattering theory

Necessary and sufficient conditions for constructing orthonormal wavelet bases

Wayne M. Lawton

J. Math. Phys. 32, 57 (1991); http://dx.doi.org/10.1063/1.529093 (5 pages) | Cited 62 times

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This paper proves a previous conjecture of the author characterizing sequences h∊l2(Z) that yield orthonormal wavelet bases of L2(R) in terms of the multiplicity of the eigenvalue 1 of an operator associated to h. The proof utilizes a result of Cohen characterizing these sequences in terms of the real zeros of their Fourier transforms. The mapping from sequences to wavelets is shown to define a continuous mapping from a subset of l2(Z) into L2(R). Related conjectures are discussed.
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02.30.-f Function theory, analysis

An extension of Szegö’s theorem

T. Morita

J. Math. Phys. 32, 62 (1991); http://dx.doi.org/10.1063/1.529094 (7 pages)

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An extension of Szegö’s theorem, Theorem 1 in the text, is proved by using the elementary theory of complex functions and the Wiener–Lévy theorem.
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02.30.-f Function theory, analysis
05.50.+q Lattice theory and statistics (Ising, Potts, etc.)

Painlevé analysis and reducibility to the canonical form for the generalized Kadomtsev–Petviashvili equation

Tommaso Brugarino and Antonio M. Greco

J. Math. Phys. 32, 69 (1991); http://dx.doi.org/10.1063/1.529095 (3 pages) | Cited 7 times

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The most general Kadomtsev–Petviashvili (KP) type equation, [ut+math(t,x,y)u+b(t,x,y) ux+c(t,x,y)uux+d(t, x,y)uxxx]x+k(t,x,y) uyy=e(t,x,y), is studied and the conditions for the coefficients, in order that it owns complete integrability, are determined via a Painlevé test. Finally, it is proved that the above conditions are the same as those requested for reducing the equation to the canonical form via suitable transformations.
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02.30.Jr Partial differential equations

Lax pairs galore

F. Calogero and M. C. Nucci

J. Math. Phys. 32, 72 (1991); http://dx.doi.org/10.1063/1.529096 (3 pages) | Cited 15 times

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A formula that yields an (apparently—but only apparently—nontrivial) Lax pair for any nonlinear evolution PDE in 1+1 dimensions possessing a local conservation law is presented. Several examples are exhibited.
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02.30.Jr Partial differential equations
41.20.Jb Electromagnetic wave propagation; radiowave propagation

Generalized Lie symmetries and complete integrability of certain nonlinear Hamiltonian systems with three degrees of freedom

M. Lakshmanan and R. Sahadevan

J. Math. Phys. 32, 75 (1991); http://dx.doi.org/10.1063/1.529097 (9 pages) | Cited 12 times

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A systematic method for investigating the existence of nontrivial generalized Lie symmetries is presented and the associated integrals of motion for nonlinear oscillator systems with three‐degrees of freedom defined in terms of the Lagrangian by L= (1)/(2) (math2+math2+math2)−V(x,y,z) are constructed. Then the method is applied to study the integrability properties of quartically and cubically coupled nonlinear oscillators with three degrees of freedom. Compatibility with the Painlevé property is also investigated.
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02.30.Jr Partial differential equations
02.20.-a Group theory

Metrics and dual operators

Gerald Harnett

J. Math. Phys. 32, 84 (1991); http://dx.doi.org/10.1063/1.529098 (8 pages) | Cited 11 times

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For a four‐dimensional vector space it is possible to define independently of a metric the notion of a dual operator acting on bivectors. It is shown that the map which takes a conformal class of metrics together with an orientation to the induced dual operator is an isomorphism. For each metric signature type, this map is equivalent to an isomorphism of certain homogeneous spaces.
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02.40.-k Geometry, differential geometry, and topology
04.20.Cv Fundamental problems and general formalism

Contraction of information in systems far from equilibrium

Kuniharu Kishida

J. Math. Phys. 32, 92 (1991); http://dx.doi.org/10.1063/1.529100 (7 pages) | Cited 4 times

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The relationship between a physical model and a time series model (or an innovation model) is discussed from the viewpoint of contraction of information due to the way of observation. In this framework, measurable correlation functions are expressed in terms of both models. By using the Riccati type equation and the singular value decomposition of the Hankel matrix, a data‐oriented model is derived from measurable correlation functions. There occurs the contraction in a noise source space (or random forces). From the mechanism of contraction, random forces of systems can be identified by using two innovation models in different states.
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02.50.-r Probability theory, stochastic processes, and statistics
06.90.+v Other topics in metrology, measurements, and laboratory procedures (restricted to new topics in section 06)
82.20.Fd Collision theories; trajectory models
82.40.Bj Oscillations, chaos, and bifurcations

The Burgers equation on the semiline with general boundary conditions at the origin

F. Calogero and S. De Lillo

J. Math. Phys. 32, 99 (1991); http://dx.doi.org/10.1063/1.529101 (7 pages) | Cited 11 times

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A technique is given to solve the initial/boundary value problem for the Burgers equation ut(x,t)=uxx(x,t)+2 ux(x,t) u(x,t) on the semiline 0≤x<∞, with the general boundary condition at the origin H[u(0,t),ux(0,t);t]=0. Here ‘‘to solve’’ means ‘‘to reduce to an equation in one variable only.’’ This equation is generally nonlinear and integrodifferent ial; it comes in several (equivalent) avatars, which contain nontrivially a free parameter, whose value can be assigned arbitrarily since the solution of the equation is independent of it. In the special case when H(y,z;t)=a(t)y+b(t)(z+y2) −F(t), which is the case relevant for most applications, the equations reduce to linear integral equations of Volterra type, which can in fact be solved by quadratures if a(t)/F(t)=c1 and b(t)/F(t)=c2 are time‐independent.
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02.60.Lj Ordinary and partial differential equations; boundary value problems

Wake‐free waves in one and three dimensions

L. Bombelli, W. E. Couch, and R. J. Torrence

J. Math. Phys. 32, 106 (1991); http://dx.doi.org/10.1063/1.529132 (3 pages) | Cited 2 times

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A recent paper by Gottlieb [J. Math. Phys. 29, 2434 (1988)] provides examples of acoustic wave equations, in various dimensions, that have nontrivial families of solutions that are progressing waves of order 1, and relates this to whether or not these equations satisfy Huygens’ principle. A statement made in that paper related to Huygens’ principle in one space dimension is clarified, and it is shown in this connection that, in general, the relationship between the possession of progressing wave solutions and the satisfaction of Huygens’ principle is more complex than the situation described by Gottlieb. In addition, the attractive properties of progressing waves of order 1 are retained by progressing waves of any finite order, and we use this to generalize in several ways Gottlieb’s results on ‘‘wake‐free’’ solutions of the acoustic equation in three dimensions.
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02.90.+p Other topics in mathematical methods in physics (restricted to new topics in section 02)
43.20.+g General linear acoustics

Classification results and the Darboux theorem for low‐order Hamiltonian operators

David B. Cooke

J. Math. Phys. 32, 109 (1991); http://dx.doi.org/10.1063/1.529133 (11 pages) | Cited 4 times

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Hamiltonian operators and their behavior under differential substitutions are studied. Scalar Hamiltonian operators are classified up to fifth order, and it is shown that all such operators may be obtained from the first‐order Gardner operator, Dx, by differential substitutions, thus proving an infinite‐dimensional Darboux theorem for Hamiltonian systems of evolution equations corresponding to such operators.
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02.90.+p Other topics in mathematical methods in physics (restricted to new topics in section 02)
02.40.Ma Global differential geometry
47.35.-i Hydrodynamic waves

Linearization of novel nonlinear diffusion equations with the Hilbert kernel and their exact solutions

Y. Matsuno

J. Math. Phys. 32, 120 (1991); http://dx.doi.org/10.1063/1.529134 (7 pages) | Cited 3 times

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Two novel nonlinear diffusion equations with the Hilbert kernel are proposed. The equations can be linearized by introducing appropriate dependent variable transformations. The initial value problems for the proposed equations are then solved exactly through the linearization and explicit nonperiodic and periodic solutions are constructed. The properties of solutions are investigated in detail. It is found that the blow up of solutions occurs at a finite time for both the nonperiodic and periodic cases due to the breakdown of certain analytic conditions imposed on the dependent variables.
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02.90.+p Other topics in mathematical methods in physics (restricted to new topics in section 02)

Redefinition of position variables and the reduction of higher‐order Lagrangians

Thibault Damour and Gerhard Schäfer

J. Math. Phys. 32, 127 (1991); http://dx.doi.org/10.1063/1.529135 (8 pages) | Cited 23 times

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Single‐time Lagrangians are treated in this paper, describing the dynamics of systems of point particles, which are given as formal power series in some ordering parameter and which may contain higher time derivatives in all terms but the leading one. An efficient method for eliminating the higher time derivatives directly on the Lagrangian level is presented. This method clarifies the meaning of using the lower‐order equations of motion in higher‐order terms in a Lagrangian. The method consists of an iterative use of ‘‘contact’’ transformations in the jet prolongation of the extended configurations space and is called ‘‘the method of redefinition of position variables.’’ Several examples from electrodynamics and relativistic gravity are treated explicitly.
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45.05.+x General theory of classical mechanics of discrete systems
03.30.+p Special relativity
03.50.-z Classical field theories
02.90.+p Other topics in mathematical methods in physics (restricted to new topics in section 02)

Phase operators via group contraction

D. Ellinas

J. Math. Phys. 32, 135 (1991); http://dx.doi.org/10.1063/1.529136 (7 pages) | Cited 32 times

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The problem of quantization of the classical phase of a harmonic oscillator (HO) is solved here in two steps. First, polar decomposition of the step operators of the u(2) algebra is performed. Second, the method of group contraction is used through which, in the limit j→∞, math,math° is passed to for the quantized HO and its Hermitian phase operators. Also, phase states, i.e., states with sharply defined phase, are constructed and the dynamical aspects of the contraction limit between the Jaynes–Cummings model (JCM) and a finite‐dimensional counterpart with increasing j parameter are studied. Finally, the old problem of the phase operators is discussed in the wider frame of rank‐1 algebras and classify the previous works in this frame.
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03.65.Ca Formalism
02.20.-a Group theory

No Lagrangian? No quantization!

Sergio A. Hojman and L. C. Shepley

J. Math. Phys. 32, 142 (1991); http://dx.doi.org/10.1063/1.529507 (5 pages) | Cited 33 times

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This work starts with classical equations of motion and sets very general quantization conditions (commutation relations). It is proved that these conditions imply that the equations of motion are equivalent to the Euler–Lagrange equations of a Lagrangian L. The result is a generalization of work by Feynman, recently reported by Dyson [Am. J. Phys. 58, 209–211 (1990)]. The Lagrangian L need not be unique. Examples are given, including classical equations that do not come from a Lagrangian and therefore cannot be quantized consistently.
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03.65.Ca Formalism
45.05.+x General theory of classical mechanics of discrete systems

p‐adic path integrals

E. I. Zelenov

J. Math. Phys. 32, 147 (1991); http://dx.doi.org/10.1063/1.529137 (6 pages) | Cited 5 times

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The definition of a path integral is proposed. The method suggested is analogous to Lagrange’s formulation of a path integral used in ordinary quantum mechanics. The notation of linear order on the set of p‐adic number, p‐adic segment, p‐adic Lagrangian, integral of p‐adic function of one variable and classical action are introduced. It is proven that if the action is stationary at some trajectory then the Euler–Lagrange equations are satisfied on this trajectory. A finite approximation of a path integral is constructed and the kernel of the operator of evolution is calculated for the case of p‐adic harmonic oscillator.
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03.65.Db Functional analytical methods
02.30.-f Function theory, analysis

Three‐body quantum system on a line: Wave function asymptotics for increasing interactions

Yu. A. Kuperin, S. P. Merkuriev, and E. A. Yarevsky

J. Math. Phys. 32, 153 (1991); http://dx.doi.org/10.1063/1.529138 (4 pages)

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The asymptotical analysis for the wave function of a three‐body quantum system on a line with increasing interactions is given within the framework of the eikonal method in configuration space. For the linear increasing interactions, an explicit solution of the eikonal equation is constructed.
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03.65.Ge Solutions of wave equations: bound states
13.40.-f Electromagnetic processes and properties

Integrable systems associated with the Schrödinger spectral problem in the plane

Y. Cheng

J. Math. Phys. 32, 157 (1991); http://dx.doi.org/10.1063/1.529139 (6 pages) | Cited 10 times

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Associated with the Schrödinger spectral problem in the plane, the nonlinear integrable systems are investigated. The commutative properties of equations of the flow and their symmetries and mastersymmetries are derived in spite of the noncanonicality of the related extended recursion operator. By applying the inverse scattering transform, the systems can be solved and then the correspondences between commuting (resp. noncommuting) flows and isospectral (resp. nonisospectral) deformations of the spectral problem are found.
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03.65.Nk Scattering theory
02.30.Jr Partial differential equations

Asymptotic behavior of Jost functions near resonance points for Wigner–von Neumann type potentials

Martin Klaus

J. Math. Phys. 32, 163 (1991); http://dx.doi.org/10.1063/1.529140 (12 pages) | Cited 9 times

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In this work are considered radial Schrödinger operators −ψ″+V(r)ψ=Eψ, where V(r)=a sin br/r+W(r) with W(r) bounded, W(r)=O(r−2) at infinity (a,b real). The asymptotic behavior of the Jost function and the scattering matrix near the resonance point E0=b2/4 are studied. If ‖a‖>‖b‖, then this point may be an eigenvalue embedded in the continuous spectrum. The leading behavior of the Jost function for all values of a and b were determined. Somewhat surprisingly, situations were found where the Jost function becomes singular as E→E0 even if E0 is an embedded eigenvalue. Moreover, it was found that the scattering matrix is always discontinuous at E0 except in a few special cases. It is also shown that the asymptotics for the Jost function and the scattering matrix hold under weaker assumptions on W(r), in particular an angular momentum term l(l+1)r−2 may be incorporated into W(r). The results were also applied to a whole line problem with a potential V(x) such that V(x)=0 for x<0 and V(x) of Wigner–von Neumann type for x>0, and the behavior of the transmission and reflection coefficients as E→E0 was also studied.
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03.65.Nk Scattering theory
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