Showing papers in "Journal of Mathematical Physics in 1992"
TL;DR: In this article, the basic properties of k-symplectic Lie algebras are introduced and developed in analogy with the well-known symplectic differential geometry, and examples of such structures related to k•symmetric differential geometry are given.
Abstract: The basic properties of k‐symplectic structures are introduced and developed in analogy with the well‐known symplectic differential geometry. Examples of such structures related to k‐symplectic Lie algebras are given.
292 citations
TL;DR: In this article, it was shown by algebraic means that this can be done, provided the extra part of the 5D geometry is used appropriately to define an effective 4D energymomentum tensor.
Abstract: Following earlier work, it is inquired how far the 5‐D Kaluza–Klein equations without sources may be reduced to the Einstein equations with sources. It is shown by algebraic means that this can be done, provided the extra part of the 5‐D geometry is used appropriately to define an effective 4‐D energy‐momentum tensor. The latter has reasonable properties, but will require further detailed study.
246 citations
TL;DR: In this paper, a k-constrained k-order polynomial coupled with a first-order one is proposed to lead the Kadomtsev-Petviashvili (KP) hierarchy to a nonlinear system involving a finite number of dynamical coordinates.
Abstract: For 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.
238 citations
TL;DR: In this article, the integrability aspects of a classical one-dimensional continuum isotropic biquadratic Heisenberg spin chain in its continuum limit up to order [O(a4)] in the lattice parameter "a" are studied.
Abstract: The integrability aspects of a classical one‐dimensional continuum isotropic biquadratic Heisenberg spin chain in its continuum limit up to order [O(a4)] in the lattice parameter ‘‘a’’ are studied. Through a differential geometric approach, the dynamical equation for the spin chain is expressed in the form of a higher‐order generalized nonlinear Schrodinger equation (GNLSE). An integrable biquadratic chain that is a deformation of the lower‐order continuum isotropic spin chain, is identified by carrying out a Painleve singularity structure analysis on the GNLSE (also through gauge analysis) and its properties are discussed briefly. For the nonintegrable chain, the perturbed soliton solution is obtained through a multiple scale analysis.
193 citations
TL;DR: Affine Wigner functions are phase space representations based on the affine group in place of the usual Weyl-Heisenberg group of quantum mechanics as mentioned in this paper, which are relevant to the time-frequency analysis of real signals.
Abstract: Affine Wigner functions are phase space representations based on the affine group in place of the usual Weyl–Heisenberg group of quantum mechanics. Such representations are relevant to the time–frequency analysis of real signals. An interesting family is singled out by the requirement of covariance with respect to each solvable three‐parameter group containing the affine group. Explicit forms are given in each case and properties such as unitarity and localization are discussed. Some particular distributions are recovered.
127 citations
TL;DR: In this article, it was shown that the only static, asymptotically flat solutions of the coupled Einstein-matter equations with regular event horizon and finite energy consist of the Schwarzschild metric and a constant map, being a zero of the non-negative potential.
Abstract: The coupled system of gravity and mappings φ:(M,g)→(N,G) with harmonic action and additional potential is considered. For spherically symmetric manifolds (M,g) and Riemannian manifolds (N,G) it is shown that the only static, asymptotically flat solutions of the coupled Einstein‐matter equations with regular event horizon and finite energy consist of the Schwarzschild metric and a constant map, being a zero of the non‐negative potential.
122 citations
TL;DR: A generalization of supersymmetry based on Z3-graded algebras was proposed in this article, where the operators whose trilinear combinations yield the supersymmetric translation generators can be constructed.
Abstract: A generalization of supersymmetry is proposed based on Z3 ‐graded algebras. Introducing the objects whose ternary commutation relations contain the cubic roots of unity, e2πi/3, e4πi/3 and 1, the operators whose trilinear combinations yield the supersymmetric translation generators can be constructed. Cubic matrices forming a ternary algebra are the generalization of Pauli’s matrices. The general properties of Z3 ‐grading, some other representations of such algebras, and their possible pertinence with regard to the quark model are briefly discussed.
121 citations
TL;DR: In this article, a generalization of self-similar approximations is proposed and stability conditions for the method are formulated: first, the mapping multipliers for the self−similar recurrence relations have to be less than unity; second, the Lyapunov exponent for the differential form of the self•similar relations has to be negative; and third, the fixed point conditions defining the governing functions are analyzed from the point of view of stability.
Abstract: A generalization for the method of self‐similar approximations suggested recently by the author is given. Stability conditions for the method are formulated: first, the mapping multipliers for the self‐similar recurrence relations have to be less than unity; second, the Lyapunov exponent for the differential form of the self‐similar relations has to be negative. The fixed‐point conditions defining the governing functions are analyzed from the point of view of stability. It is shown that the fixed‐point condition expressed in the form of the principle of minimal sensitivity provides a contracting mapping contrary to the condition in the form of the principle of minimal difference. This explains why, in general, the former fixed‐point condition yields more accurate results than the latter.
107 citations
TL;DR: In this article, general expressions for scalar one-loop massive Feynman diagrams with any number of external lines and with arbitrary momenta of these lines, with arbitrary values of masses and powers of denominators of internal lines, the space-time dimension n also being arbitrary.
Abstract: General expressions are obtained for scalar one‐loop massive Feynman diagrams with any number of external lines and with arbitrary momenta of these lines, with arbitrary values of masses and powers of denominators of internal lines, the space‐time dimension n also being arbitrary. Special cases are considered when two (or more) masses are equal to 0. The results are represented in terms of hypergeometric functions.
106 citations
TL;DR: In this paper, a set of coupled higher-order nonlinear Schrodinger equations, which describe electromagnetic pulse propagation in coupled optical waveguides, is formulated in terms of an eigenvalue problem.
Abstract: A set of coupled higher‐order nonlinear Schrodinger equations, which describe electromagnetic pulse propagation in coupled optical waveguides, is formulated in terms of an eigenvalue problem. Using that result, the inverse scattering problem is solved and explicit soliton solutions are found. Additionally, linear coupling terms are studied systematically.
105 citations
TL;DR: In this article, a method of constructing Lax representations for isospectral and nonispectral hierarchies of evolution equations is proposed, and it is shown that under some simple but essential conditions, the corresponding Lax operators may constitute infinite-dimensional Lie algebras with respect to the binary operation.
Abstract: A method of constructing Lax representations for isospectral and nonisospectral hierarchies of evolution equations is proposed. It is shown that under some simple but essential conditions, the corresponding Lax operators may constitute infinite‐dimensional Lie algebras with respect to the binary operation ■⋅,⋅■ given in this paper. Furthermore, the detailed analyses for the KdV couple, the AKNS couple, and a new couple of isospectral and nonisospectral hierarchies of integrable equations are presented as examples.
TL;DR: In this article, the authors considered the inverse scattering problem for a potential V(x) supported on the half-line when the given data is R−(k) and the amplitude of the reflection coefficient was determined.
Abstract: Two related problems are considered: (i) the inverse scattering problem for a potential V(x) supported on the half‐line {x≥0}, when the given data is ‖R−(k)‖, the amplitude of the reflection coefficient and (ii) determination of a function g(t) supported on the half‐line {t≥0} when the given data is ‖g(k)‖, the amplitude of the Fourier transform of g. Under certain conditions on V or g, uniqueness theorems are proved and computational methods are developed. A numerical example of recovery of V(x) from ‖R−(k)‖ is given.
TL;DR: In this article, the integrable coordinate systems with hyperbolic Gaudin magnet are considered, and an isomorphism is given for these systems with the same magnet.
Abstract: The integrable systems are considered which are connected with separation of variables in real Riemannian spaces of constant curvature. An isomorphism is given for these systems with hyperbolic Gaudin magnet. Using this isomorphism, the complete classification of separable coordinate systems is provided by means of the corresponding L‐operators for the Gaudin magnet.
TL;DR: In this paper, Sato's approach is applied to the Kadomtsev-Petviashvili (KP) hierarchy and the two-dimensional Toda lattice (2DTL) hierarchy.
Abstract: New 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 u1,u2,u3,..., 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=1N αnSn(u1,u2,u3,...)=Ωx, where Sn are symmetries for these hierarchies, αn 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 u2,u3,...,uN,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 hierarc...
TL;DR: In this article, a finite Euler hierarchy of field theory Lagrangians leading to universal equations of motion for new types of string and membrane theories and for classical topological field theories is constructed.
Abstract: Finite Euler hierarchies of field theory Lagrangians leading to universal equations of motion for new types of string and membrane theories and for classical topological field theories are constructed. The analysis uses two main ingredients. On the one hand, there exists a generic finite Euler hierarchy for one field leading to a universal equation which generalizes the Plebanski equation of self‐dual four‐dimensional gravity. On the other hand, specific maps are introduced between field theories which provide a ‘‘triality’’ between certain classes of arbitrary field theories, classical topological field theories and generalized string and membrane theories. The universal equations, which derive from an infinity of inequivalent Lagrangians, are generalizations of certain reductions of the Plebanski and KdV equations, and could possibly define new integrable systems, thus in particular integrable membrane theories. Some classes of solutions are constructed in the general case. The general solution to some ...
TL;DR: In this article, a supersymmetric generalization of the cubic Schrodinger equation is proposed, which admits an infinite set of (higher-order) local and nonlocal symmetries and conservation laws, the lowest order terms of which are given explicitly.
Abstract: A construction is proposed for a supersymmetric generalization of the cubic Schrodinger equation, resulting in two supersymmetric systems, one of which contains a free parameter. Both systems are proven to admit an infinite set of (higher‐order) local and nonlocal symmetries and a seemingly infinite set of conservation laws, the lowest‐order terms of which are given explicitly. Moreover, the theory of coverings (equivalent to the prolongation method of Wahlquist and Estabrook) is applied to both systems. Both are seen to admit an infinite‐dimensional covering algebra, the structure of which is determined explicitly, resulting in a related supersymmetric system of differential equations, as well as an auto‐Backlund transformation for each equation. This indicates the complete integrability of both systems.
TL;DR: In this paper, it was shown that for certain equations of state there exists a wide class of solutions of this type corresponding to appropriate initial data given on a spacelike hypersurface.
Abstract: A body or collection of bodies made of perfect fluid can be described in general relativity by a solution of the Einstein–Euler system where the mass density has spatially compact support. It is shown that for certain equations of state there exists a wide class of solutions of this type corresponding to appropriate initial data given on a spacelike hypersurface. This class is not constrained by any symmetry requirements. The key element of the proof is to write the equations as a symmetric hyperbolic system which is regular both for nonvanishing density and in vacuum.
TL;DR: In this article, the symmetry group of the KdV equation is shown to be at most four dimensional, and this occurs if and only if the equation is equivalent, under local point transformations, to the kdV with f = g = 1.
Abstract: The Lie point symmetries of the equation ut+f(x, t)uux+g(x,t)uxxx=0 are studied. The symmetry group is shown to be, at most, four dimensional, and this occurs if and only if the equation is equivalent, under local point transformations, to the KdV equation with f=g=1. For nine different classes of functions f and g, the symmetry group turns out to be three dimensional. Two‐dimensional and one‐dimensional symmetry groups occur for 11 and 15 classes of equations, respectively.
TL;DR: In this article, the case of a matter bundle E whose standard fiber admits action only of an exact symmetry subgroup H of G is examined, where the bundle E fails to be associated with a principal bundle and the canonical jet bundle morphism J1EH×J1Σ→J1E is used.
Abstract: Given a principal bundle P→X with a structure group G and an associated Higgs bundle Σ with a standard fiber G/H, the case of a matter bundle E whose standard fiber admits action only of an exact symmetry subgroup H of G is examined. In the presence of a fixed Higgs field σ: X→Σ, matter fields are represented by sections of a matter bundle Eh associated with the corresponding reduced subbundle Ph of P. The totality of matter fields and Higgs fields is described by sections of the bundle E which is the composite bundle EH→Σ→X where EH→Σ is the bundle associated with the principal H‐bundle P→Σ. The bundle E fails to be associated with a principal bundle. To construct a connection Γ: E→J1E on E, the canonical jet bundle morphism J1EH×J1Σ→J1E is used.
TL;DR: In this article, the Stratonovich-Weyl kernels are constructed for some of the coadjoint orbits of the two-dimensional extended Galilean group G(2+1) and the unitary irreducible representations associated with a given group orbit are obtained using the Kirillov-Mackey theory.
Abstract: Stratonovich–Weyl kernels are constructed for some of the coadjoint orbits of the two‐dimensional extended Galilean group G(2+1). As an intermediate step, the unitary irreducible representations associated with a given group orbit are obtained by using the Kirillov–Mackey theory. Star products are defined, in the sense of Moyal, for functions on each of these orbits. The central extension of G(2+1) with parameter k is also analyzed, which results from the commutator between the generators of boosts, to conclude that it originates a sort of nonrelativistic remainder of the Thomas precession.
TL;DR: In this paper, a subgroup corresponding to axisymmetric geometry is chosen, and the details of the construction of the one-and two-dimensional optimal systems are shown.
Abstract: The group properties of the three‐dimensional (3‐D), one‐temperature hydrodynamic equations, including nonlinear conduction and a thermal source, are presented. A subgroup corresponding to axisymmetric geometry is chosen, and the details of the construction of the one‐ and two‐dimensional optimal systems are shown. The two‐dimensional optimal system is used to generate 23 intrinsically different reductions of the 2‐D partial differential equations to ordinary differential equations. These ordinary differential equations can be solved to provide analytic solutions to the original partial differential equations. Two example analytic solutions are presented: a 2‐D axisymmetric flow with a P2 asymmetry and a 3‐D spiraling flow.
TL;DR: In this paper, it was shown that Up,q dual to GLp,q(2,C) is isomorphic to U(pq)1/2(sl( 2,C)) ⊗ Z as a commutation algebra, where Z is a subalgebra central in up,q.
Abstract: It is shown that the algebra Up,q dual to GLp,q(2,C) is isomorphic to U(pq)1/2(sl(2,C)) ⊗ Z as a commutation algebra, where Z is a subalgebra central in Up,q. The subalgebra Z is a Hopf subalgebra of Up,q, while the commutation subalgebra U(pq)1/2(sl(2,C)) is not a Hopf subalgebra.
TL;DR: In this paper, the conformal invariance properties of a Dirac oscillator are established and a set of operators whose algebra shows that it can be considered as a conformal system is constructed.
Abstract: The conformal invariance properties of a Dirac oscillator are established. A set of operators is constructed whose algebra shows that it can be considered as a conformal system. The operators are then used to solve the problem using algebraic techniques. The superconformal generalization of the algebra is also worked out, and some consequences of these invariances for the properties of the model are mentioned.
TL;DR: In this article, a method of constructing finite-dimensional integrable systems starting from a bi-Hamiltonian hierarchy of soliton equations is introduced, where the existence of two Hamiltonian structures of the hierarchy leads to a bi−Hamiltonian formulation of the resulting finite−dimensional systems.
Abstract: A systematic method of constructing finite‐dimensional integrable systems starting from a bi‐Hamiltonian hierarchy of soliton equations is introduced. The existence of two Hamiltonian structures of the hierarchy leads to a bi‐Hamiltonian formulation of the resulting finite‐dimensional systems. The case of coupled KdV hierarchies is studied in detail. A surprising connection with separable Jacobi potentials is uncovered and described.
TL;DR: In this article, the moduli space metric for hyperbolic vortices is constructed, and their slow motion scattering is calculated in terms of geodesics in this space, which is the same metric used in this paper.
Abstract: The moduli space metric for hyperbolic vortices is constructed, and their slow motion scattering is calculated in terms of geodesics in this space.
TL;DR: In this article, the modified Carleman embedding method is used to find invariants of the motion of polynomial form to the Lotka-Volterra system.
Abstract: The modified Carleman embedding method already introduced by the authors to find first integrals (invariants of the motion) of polynomial form to the Lotka–Volterra system is described in detail, and its efficiency to treat the N‐dimensional system proved. Using this method, an extensive investigation is performed for polynomials of the first degree, which allow a classification of the integrals in three families. For some systems possessing one invariant it is possible to find a second invariant using rescaling methods. They represent very restrictive solutions, implying that there exists a great number of conditions among the equation’s coefficients to satisfy. A proof is given that the Volterra invariants can be deduced as a limit. Finally, the interesting properties of the solutions of these systems are studied in detail.
TL;DR: In this article, the spectral representations of the fixed energy amplitude of the symmetric and general Poschl-Teller potentials are summed via a Sommerfeld-Watson transformation which leads to a simple closed-form expression.
Abstract: The spectral representations of the fixed‐energy amplitude of the symmetric and the general Poschl–Teller potentials are summed via a Sommerfeld–Watson transformation which leads to a simple closed‐form expression. The result is used to write down a similar expression for the symmetric and general Rosen–Morse potentials, exploiting the close correspondence that exists between the two systems within the Schrodinger theory and the path integral formalism via a Duru–Kleinert transformation. From the singularities of the latter amplitude the bound and continuum states of the general Rosen–Morse potential are extracted.
TL;DR: In this paper, various forms of the nonlinear Klein-Gordon equation are seen to have exact soliton-like solutions when separation of variables is postulated, and the family for which these exact solutions are found includes the sine−Gordon equation as a special case.
Abstract: Various forms of the nonlinear Klein–Gordon equation are seen to have exact, solitonlike solutions when separation of variables is postulated. The family for which these exact solutions are found includes the sine‐Gordon equation as a special case. An interesting conclusion obtained is that both the soliton–antisoliton and breather solutions, hitherto known for the sine‐Gordon equations result from a broader class of Klein–Gordon equations. The method can be extended to other equations.
TL;DR: Under some natural assumptions [less restrictive than in the paper by Wess and Zumino (preprint CERN−TH•5697/90, LAPP•TH•284/90)] differential calculi on the quantum plane are found and investigated as mentioned in this paper.
Abstract: Under some natural assumptions [less restrictive than in the paper by Wess and Zumino (preprint CERN‐TH‐5697/90, LAPP‐TH‐284/90)] differential calculi on the quantum plane are found and investigated. Complex structure, complex derivatives, and holomorphic functions on the quantum plane are defined. Generalized Cauchy–Riemann equations are given and solved.
TL;DR: In this paper, a starquantization of the Klein-Gordon equation based on Poincare group is discussed and several examples are discussed and the work is finished with some remarks on perturbation theory induced by a starproduct.
Abstract: A study of star‐quantization of the Klein–Gordon equation based on Poincare group is presented. Several examples are discussed and the work is finished with some remarks on perturbation theory induced by a star‐product.