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Volume 25, Issue 12, December 1984

Properties of Poincaré generating functions for polynomial covariants (tensors)
View Description Hide DescriptionWe 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.

Lie groups and Lie algebras with generalized supersymmetric parameters
View Description Hide DescriptionMatrices 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.

Determination of point group harmonics for arbitrary j by a projection method. III. Cubic group, quantization along a ternary axis
View Description Hide DescriptionThe 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 D _{3}. Relations between the descriptions on the two axes of quantization are derived.

A nilpotent prolongation of the Robinson–Trautman equation
View Description Hide DescriptionA 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.

Factorization method and new potentials with the oscillator spectrum
View Description Hide DescriptionA 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.

The double cnoidal wave of the Korteweg–de Vries equation: An overview
View Description Hide DescriptionEarlier 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.

Perturbation series for the double cnoidal wave of the Korteweg–de Vries equation
View Description Hide DescriptionBy 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 c _{1} and c _{2}. 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.

The special modular transformation for polycnoidal waves of the Korteweg–de Vries equation
View Description Hide DescriptionThe 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.

On an extension of the classical Thirring model
View Description Hide DescriptionA new class of classical field theories, in 1+1 dimensions, is introduced, of the form iγ^{μ}Ψ_{, μ}−mΨ−Ψ̄γ^{μ}(g _{1}+g _{2}γ^{5}) ×Ψγ_{μ}Ψ−Ψ̄(g _{3}+g _{4}γ^{5})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 g _{3} and g _{4} 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.

Landau–Lifshitz and higher‐order nonlinear systems gauge generated from nonlinear Schrödinger‐type equations
View Description Hide DescriptionNew 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.

Generalized logarithmic Borel summability
View Description Hide DescriptionThe 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.

On the solutions to a class of nonlinear integral equations arising in transport theory
View Description Hide DescriptionExistence 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.

Two theorems on star diagrams
View Description Hide DescriptionThe 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.

Formulas for the eigenvalues of the Laplacian on tensor harmonics on symmetric coset spaces
View Description Hide DescriptionOn 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 S _{ n }, C P _{ n }, and real (complex) Grassmann manifolds are analyzed.

A certain class of solutions of the nonlinear wave equation
View Description Hide DescriptionIn this paper are investigated some differential geometry methods in the theory of the nonlinear waveequation ∇^{2} u=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=4A v ^{3}+3B v ^{2}+2c v+D, ⧠v=μ exp v, ⧠v=sin v, ⧠v=cosh v, a n d ⧠v=sinh v.

Hamiltonians with high‐order integrals and the ‘‘weak‐Painlevé’’ concept
View Description Hide DescriptionWe 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.

Polynomial constants of motion in flat space
View Description Hide DescriptionSome 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.

A viewpoint of Kaluza–Klein type in elasticity theory
View Description Hide DescriptionAn N‐dimensional anisotropicelastic 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 anisotropicelastic bodies includes Bromwich’s method of solving the equation of motion of incompressible isotropic elastic bodies under the influence of the interior gravity.

Multifrequency inverse problem for the reduced wave equation: Resolution cell and stability
View Description Hide DescriptionThe multifrequency inverse problem associated with the reduced wave equation Δu+k ^{2} n ^{2}(x)u=0, x ∈ R ^{3} 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.

A rule for the total number of topologically distinct Feynman diagrams
View Description Hide DescriptionA 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.