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Dec 1984

Volume 25, Issue 12, pp. 3363-3563

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Properties of Poincaré generating functions for polynomial covariants (tensors)

Marko V. Jarić

J. Math. Phys. 25, 3363 (1984); http://dx.doi.org/10.1063/1.526104 (4 pages) | Cited 1 time

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We show that irreducible polynomial covariants (tensors) based on a given irreducible representation of a finite group must have a definite p‐phase, i.e., degree modulo p, where p is the order of the center of the image of the group. We also show that the numerator of the Poincaré function is often a polynomial symmetric around a given degree and we derive several interesting properties of the Poincaré function.
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02.20.-a Group theory
02.30.-f Function theory, analysis
11.15.-q Gauge field theories
61.50.Ah Theory of crystal structure, crystal symmetry; calculations and modeling

Lie groups and Lie algebras with generalized supersymmetric parameters

Yuji Kobayashi and Shigeaki Nagamachi

J. Math. Phys. 25, 3367 (1984); http://dx.doi.org/10.1063/1.526105 (8 pages) | Cited 10 times

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Matrices with σ‐symmetric parameters (the most general extension of supersymmetric parameters) are investigated. The superdeterminants of such matrices are defined. Lie groups consisting of these matrices and their Lie algebras are studied.
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02.20.-a Group theory
11.30.Pb Supersymmetry

Determination of point group harmonics for arbitrary j by a projection method. III. Cubic group, quantization along a ternary axis

Jacques Raynal and Robert Conte

J. Math. Phys. 25, 3375 (1984); http://dx.doi.org/10.1063/1.526106 (7 pages) | Cited 3 times

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The method described in a first paper to obtain cubic harmonics quantized on an axis of order 4 is applied to the case of a ternary quantization axis. Projectors on irreducible representations are expressed with rotation matrices R(0, φ, π) and R(π, π‐φ, 0), φ=arccos(1/3), acting on subspaces of SU(2) invariant under D3. Relations between the descriptions on the two axes of quantization are derived.
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02.20.-a Group theory
31.15.-p Calculations and mathematical techniques in atomic and molecular physics
61.50.Ah Theory of crystal structure, crystal symmetry; calculations and modeling

A nilpotent prolongation of the Robinson–Trautman equation

E. N. Glass and D. C. Robinson

J. Math. Phys. 25, 3382 (1984); http://dx.doi.org/10.1063/1.526107 (5 pages) | Cited 7 times

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A prolongation is constructed, in the sense of Wahlquist and Estabrook, for the nonlinear evolution equation determining Robinson–Trautman space‐times. The Lie algebra so obtained is found to be (naturally) seven‐dimensional and nilpotent. Representations of the algebra are considered. The simple relationship of such a prolongation to the conservation laws associated with the Robinson–Trautman equation is discussed.
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02.30.-f Function theory, analysis
04.20.Jb Exact solutions

Factorization method and new potentials with the oscillator spectrum

Bogdan Mielnik

J. Math. Phys. 25, 3387 (1984); http://dx.doi.org/10.1063/1.526108 (3 pages) | Cited 139 times

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A one‐parameter family of potentials in one dimension is constructed with the energy spectrum coinciding with that of the harmonic oscillator. This is a new derivation of a class of potentials previously obtained by Abraham and Moses with the help of the Gelfand–Levitan formalism.
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02.30.Em Potential theory
03.65.-w Quantum mechanics

The double cnoidal wave of the Korteweg–de Vries equation: An overview

John P. Boyd

J. Math. Phys. 25, 3390 (1984); http://dx.doi.org/10.1063/1.526109 (12 pages) | Cited 14 times

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Earlier work of the author on the spatially periodic solutions of the Korteweg–de Vries equation is here extended via an in‐depth treatment of a special case. The double cnoidal wave is the simplest generalization of the ordinary cnoidal wave discovered by Korteweg and de Vries in 1895. In the limit of small amplitude, the double cnoidal wave is the sum of two noninteracting linear sine waves. In the oppositie limit of large amplitude, it is the sum of solitary waves of two different heights repeated periodically over all space. Although special, the double cnoidal wave is important because it is but the particular case N=2 of a broad family of solutions known variously as ‘‘N‐polycnoidal waves,’’ ‘‘finite gap,’’ ‘‘finite zone’’ solutions, ‘‘waves on a circle,’’ or ‘‘N‐phase wave trains.’’ It has been shown by others that the set of N‐polycnoidal waves gives the general initial value solution to the Korteweg–de Vries equation. This present work is the core of a three‐part treatment of the double cnoidal wave. This part, the overview, presents graphic examples in all the important parameter regimes, explains how collision phase shifts alter the average speed of the two wave phases from the ‘‘free’’ velocities of the two solitary waves, describes the different branches or modes of the double cnoidal wave (it is possible to have many solitary waves on each spatial period provided they are of only two distinct sizes), and contrasts the results of this work with the very limited numerical calculations of previous authors. The second part describes how the problem of numerically calculating the double cnoidal wave can be reduced down to solving four algebraic equations by perturbation theory. The third part explains how the so‐called ‘‘modular transformation’’ of the Riemann theta functions is important in interpreting N‐polycnoidal waves.
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02.30.Jr Partial differential equations
02.60.Lj Ordinary and partial differential equations; boundary value problems

Perturbation series for the double cnoidal wave of the Korteweg–de Vries equation

John P. Boyd

J. Math. Phys. 25, 3402 (1984); http://dx.doi.org/10.1063/1.526110 (13 pages) | Cited 8 times

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By means of the theorems proved earlier by the author, the problem of the double cnoidal wave of the Korteweg–de Vries equation is reduced to four algebraic equations in four unknowns. Two of the unknowns are the nonlinear phase speeds c1 and c2. Another is a physically irrelevant integration constant. The fourth unknown is the off‐diagonal element of the symmetric, 2×2 theta matrix, which in turn gives the explicit coefficients of the Riemann theta function. The double cnoidal wave u(x,t) is then obtained by taking the second x‐derivative of the logarithm of the theta function. Two separate forms of these four nonlinear ‘‘residual’’ equations are given. One is obtained from the Fourier series of the theta function and is useful for small wave amplitude. The other is based on the Gaussian series of the theta function and is highly efficient in the large amplitude regime where the double cnoidal wave is the sum of two solitary waves. Both sets of residual equations can be solved via perturbation theory and results are given to fourth order in the Fourier case and second order in the Gaussian case. The Gaussian‐based perturbation series has the remarkable property that it converges more and more rapidly as the wave amplitude increases; the zeroth‐order solution is the familiar double solitary wave. Numerical comparisons show that the two complementary perturbation series give accurate results in all the important regions of parameter space. (The ‘‘unimportant’’ regions are those in which the double cnoidal wave is an ordinary cnoidal wave subject to a very weak perturbation.) This is turn implies that even for moderate wave amplitude where the nonlinear interactions are not weak, and yet the solitary wave peaks are not well separated, at least to the eye, it is still qualitatively legitimate to describe the double cnoidal wave as either the sum of two sine waves or of two solitary waves of different heights.
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02.30.Jr Partial differential equations
02.60.Lj Ordinary and partial differential equations; boundary value problems
02.30.Mv Approximations and expansions

The special modular transformation for polycnoidal waves of the Korteweg–de Vries equation

John P. Boyd

J. Math. Phys. 25, 3415 (1984); http://dx.doi.org/10.1063/1.526111 (9 pages) | Cited 7 times

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The modular transformation of the Riemann theta function is used to show that the implicit dispersion relation for the N‐polycnoidal waves of the Korteweg–de Vries equation has a countable infinity of branches for N≥2. Although the transformation also implies that each branch or mode can be written in a countable infinity of ways, it is also shown that there is a unique ‘‘physical’’ representation for each mode such that the parameters of the theta function can be interpreted as wavenumbers and amplitudes in the limit of either very small or very large amplitude. Unfortunately, the small amplitude ‘‘physical’’ representation is different (by a modular transformation) from the large amplitude ‘‘physical’’ representation for a given mode, but this difference explains an apparent paradox as described in the text. The general modular transformation expresses the theta function in terms of complex wavenumbers, phase speeds, and coordinates that have no physical relevance to the Korteweg–de Vries equation, but it is shown that for N≥2, there is a subgroup, here dubbed the ‘‘special modular transformation,’’ which gives a real result. This subgroup is explicitly constructed for general N and presented as a table for N=2.
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02.30.Jr Partial differential equations
02.60.Lj Ordinary and partial differential equations; boundary value problems
02.30.Uu Integral transforms
02.30.Vv Operational calculus

On an extension of the classical Thirring model

Daniel David

J. Math. Phys. 25, 3424 (1984); http://dx.doi.org/10.1063/1.526112 (9 pages) | Cited 3 times

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A new class of classical field theories, in 1+1 dimensions, is introduced, of the form iγμΨ, μ−mΨ−Ψγμ(g1+g2γ5) ×ΨγμΨ−Ψ(g3+g4γ5)ΨΨ=0. It is shown that these theories are relativistically invariant; they do not, however, preserve parity in general, and thus could be used to describe the dynamics of weak interaction processes. The prolongation structure method is used to investigate the existence of pseudopotentials. When the coupling constants g3 and g4 are zero, the corresponding theory is then characterized by an infinite family of conservation laws and is thus completely integrable. For this very case, the Bäcklund map (pseudopotential) furnishes the equivalent of a Lax pair of operators as well as a nontrivial Bäcklund transformation and solutions of soliton type.
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02.30.Jr Partial differential equations
03.50.Kk Other special classical field theories

Landau–Lifshitz and higher‐order nonlinear systems gauge generated from nonlinear Schrödinger‐type equations

A. Kundu

J. Math. Phys. 25, 3433 (1984); http://dx.doi.org/10.1063/1.526113 (6 pages) | Cited 78 times

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New Landau–Lifshitz (LL) and higher‐order nonlinear systems gauge generated from nonlinear Schrödinger (NS) type equations are presented. The consequences of gauge equivalence between different dynamical systems are discussed. The gauge connections among various LL and NS equations are found and depicted through a schematic representation.
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02.30.Jr Partial differential equations
11.15.-q Gauge field theories
11.10.Lm Nonlinear or nonlocal theories and models
02.20.-a Group theory

Generalized logarithmic Borel summability

V. Grecchi and M. Maioli

J. Math. Phys. 25, 3439 (1984); http://dx.doi.org/10.1063/1.526098 (5 pages) | Cited 4 times

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The recently introduced logarithmic Borel summation method is able to sum strongly divergent series of a particular type. A satisfactory extension to the applicability of this method, obtained by using the classical Borel–Le Roy transform, is presented. As examples we consider a class of nonpolynomial anharmonic oscillator models in the ’t Hooft simplified form.
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02.30.Lt Sequences, series, and summability

On the solutions to a class of nonlinear integral equations arising in transport theory

G. Spiga, R. L. Bowden, and V. C. Boffi

J. Math. Phys. 25, 3444 (1984); http://dx.doi.org/10.1063/1.526099 (7 pages) | Cited 4 times

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Existence and uniqueness for the solutions to a class of nonlinear equations arising in transport theory are investigated in terms of a real parameter α which can take on positive and negative values. On the basis of contraction mapping and positivity properties of the relevant nonlinear operator, iteration schemes are proposed, and their convergence, either pointwise or in norm, is studied.
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02.30.Rz Integral equations
05.60.-k Transport processes

Two theorems on star diagrams

J. C. Houard and M. Irac‐Astaud

J. Math. Phys. 25, 3451 (1984); http://dx.doi.org/10.1063/1.526100 (4 pages)

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The notion of star diagram, previously introduced for the study of Green functions of nonlinear differential operators is formulated in an algebraic frame. Two theorems are presented which make the structure of these functions explicit.
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02.30.Tb Operator theory
02.30.Hq Ordinary differential equations

Formulas for the eigenvalues of the Laplacian on tensor harmonics on symmetric coset spaces

K. Pilch and A. N. Schellekens

J. Math. Phys. 25, 3455 (1984); http://dx.doi.org/10.1063/1.526101 (5 pages) | Cited 8 times

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On a symmetric coset space G/H the eigenvalues of the Laplacian and the Lichnerowicz operator acting on arbitrary tensor harmonics are given in terms of the eigenvalues of the quadratic Casimir operators of G and H. Explicit examples for Sn, CPn, and real (complex) Grassmann manifolds are analyzed.
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02.30.Tb Operator theory

A certain class of solutions of the nonlinear wave equation

Grzegorz Cieciura and Alfred Grundland

J. Math. Phys. 25, 3460 (1984); http://dx.doi.org/10.1063/1.526102 (10 pages) | Cited 18 times

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In this paper are investigated some differential geometry methods in the theory of the nonlinear wave equation ∇2u=Φ(u,(∇u‖∇u)). A special class of solutions is discussed for which (∇u‖∇u) is constant on each level of the function u. It is proved that levels of such solutions form in the space of independent variable’s hypersurfaces with all principal curvatures constant. The general form of such hypersurfaces is given. Then it is proved that via the method of characteristics it is possible to construct (in principle) all the solutions of the discussed class. They may be obtained by integration of an ODE of second order using a special class of the polynomial functions. Some new solutions are given for equations☒v=4Av3+3Bv2+2cv+D, ☒v=μ exp v, ☒v=sin v, ☒v=cosh v, and ☒v=sinh v.
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02.40.-k Geometry, differential geometry, and topology
02.30.Jr Partial differential equations

Hamiltonians with high‐order integrals and the ‘‘weak‐Painlevé’’ concept

B. Grammaticos, B. Dorizzi, and A. Ramani

J. Math. Phys. 25, 3470 (1984); http://dx.doi.org/10.1063/1.526103 (4 pages) | Cited 24 times

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We examine the singularity structure of the equations of motion associated to integrable two‐dimensional Hamiltonians with second integrals of order higher than 2. We show in these specific examples that the integrability is associated to a singularity expansion of the ‘‘weak‐Painlevé’’ type. New cases of integrability are discovered, with still higher‐order integrals which are explicitly computed.
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45.05.+x General theory of classical mechanics of discrete systems
02.30.-f Function theory, analysis

Polynomial constants of motion in flat space

Gerard Thompson

J. Math. Phys. 25, 3474 (1984); http://dx.doi.org/10.1063/1.526114 (5 pages) | Cited 23 times

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Some general results on commuting integrals for a Hamiltonian system are given. The question of the existence of integrals which are polynomial in the momenta is investigated and the results applied to a variety of mechanical systems.
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45.05.+x General theory of classical mechanics of discrete systems
02.30.-f Function theory, analysis

A viewpoint of Kaluza–Klein type in elasticity theory

Tsunehiro Obata and Jiro Chiba

J. Math. Phys. 25, 3479 (1984); http://dx.doi.org/10.1063/1.526115 (4 pages)

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An N‐dimensional anisotropic elastic body without the interior gravity is, under some conditions concerning the Nth dimension, equivalent to an (N−1)‐dimensional isotropic elastic body under the influence of the interior gravity. According to this theorem, our method of solving the equation of free motion of anisotropic elastic bodies includes Bromwich’s method of solving the equation of motion of incompressible isotropic elastic bodies under the influence of the interior gravity.
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46.25.Cc Theoretical studies
04.20.-q Classical general relativity

Multifrequency inverse problem for the reduced wave equation: Resolution cell and stability

V. H. Weston

J. Math. Phys. 25, 3483 (1984); http://dx.doi.org/10.1063/1.526116 (6 pages) | Cited 3 times

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The multifrequency inverse problem associated with the reduced wave equation Δu+k2n2(x)u=0, x ∊ R3 is examined for the case where the data set is sparse. The resolution cell or solution set is examined in detail and is shown to be an infinite‐dimensional manifold. The concept of stability is introduced. It is shown that the intrinsic condition of structural stability to the inverse process selects out a preferred set of solutions from the solution set. The structural stability of various iterative schemes used in the inverse process are examined.
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41.20.Jb Electromagnetic wave propagation; radiowave propagation

A rule for the total number of topologically distinct Feynman diagrams

Franco Battaglia and Thomas F. George

J. Math. Phys. 25, 3489 (1984); http://dx.doi.org/10.1063/1.526117 (3 pages) | Cited 4 times

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See Also: Erratum

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A rule for the total number of topologically distinct Feynman diagrams is presented for the ground state of a system of many identical particles interacting via a two‐body potential.
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03.65.Db Functional analytical methods
02.40.-k Geometry, differential geometry, and topology

Summation of strongly divergent perturbation series

Gustavo A. Arteca, Francisco M. Fernández, and Eduardo A. Castro

J. Math. Phys. 25, 3492 (1984); http://dx.doi.org/10.1063/1.526118 (5 pages) | Cited 17 times

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A new method for summing strongly divergent perturbation series is presented. It is based on the change of the power series into a convergent sequence by means of an order‐dependent mapping obtained from a simple scaling relation. The perturbation expansions for a one‐dimensional integral and for the ground states of the anharmonic oscillator and of the linear confining potential model are accurately summed in the most unfavorable strong‐coupling limit.
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03.65.Ge Solutions of wave equations: bound states
02.30.Lt Sequences, series, and summability

Bound state energy eigenvalues for a general class of one‐dimensional problems on the whole axis (−∞, ∞) via the Prüfer transformation

I. Úlehla and D. Adamová

J. Math. Phys. 25, 3497 (1984); http://dx.doi.org/10.1063/1.526119 (6 pages)

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An extension of the Prüfer phase function method for the bound state energy calculation is presented. It is applicable to one‐dimensional problems described by the Schrödingere quation on the whole axis (−∞,∞) with a general class of potentials. Theorems are given which are a generalization of the analogous ones concerning the half‐axis (0,∞) problems that have been presented in previous papers. The method is suitable especially for numerical calculations of the bound state energy eignevalues.
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03.65.Ge Solutions of wave equations: bound states

On charges of massless particles

Jan T. Łopuszański

J. Math. Phys. 25, 3503 (1984); http://dx.doi.org/10.1063/1.526120 (7 pages) | Cited 4 times

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We are concerned with the theorem of Weinberg and Witten stating that a massless particle of helicity ‖h‖> 1/2 cannot be a carrier of a charge of an internal symmetry induced by a Lorentz covariant current and that for a massless field theory of ‖h‖>1 a Lorentz covariant energy momentum tensor cannot be constructed. We complete the proof of the theorem, as that given by Weinberg and Witten it is inconclusive. We suggest how to evade the difficulties which arise as a consequence of this theorem.
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03.70.+k Theory of quantized fields
11.10.-z Field theory
11.40.Dw General theory of currents
11.30.Cp Lorentz and Poincaré invariance

Self‐gravitating fluids of class one with nonvanishing Weyl tensor

Y. K. Gupta, S. P. Sharma, and R. S. Gupta

J. Math. Phys. 25, 3510 (1984); http://dx.doi.org/10.1063/1.526121 (3 pages) | Cited 4 times

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All the Zeldovich fluids of imbedding class one with nonvanishing Weyl tensor have been obtained by solving equations of continuity and equations of motion. All of them are found to be irrotational and therefore can be termed as self‐gravitating fluids with pressure equal to energy density.
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04.20.Jb Exact solutions
95.30.Sf Relativity and gravitation
98.80.Cq Particle-theory and field-theory models of the early Universe (including cosmic pancakes, cosmic strings, chaotic phenomena, inflationary universe, etc.)

Constraints on the nature of inertial motion arising from the universality of free fall and the conformal causal structure of space‐time

Robert Alan Coleman and Herbert Korte

J. Math. Phys. 25, 3513 (1984); http://dx.doi.org/10.1063/1.526122 (14 pages) | Cited 12 times

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According to the principle of the universality of free fall, the motions of all neutral monopole particles are governed by one common path structure. This principle does not, however, require the path structure to be geodesic; that is, the path structure need not be a projective structure. It is shown that any equation of motion structure (either a curve or a path structure) that has sufficient microisotropy to be compatible with the conformal causal structure of space‐time must be geodesic and must be unique. Hence, the empirically well‐supported principles of conformal causality and of the universality of free fall together require the existence of a unique Weyl structure on space‐time.
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04.20.-q Classical general relativity
04.20.Cv Fundamental problems and general formalism
02.40.Ky Riemannian geometries
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