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Showing papers in "Journal of Mathematical Physics in 1988"


Journal ArticleDOI
TL;DR: In this paper, it was shown that among a family of supersymmetric extensions of the Kortewegde Vries equation, there is a special system that has an infinite number of conservation laws, which can be formulated in the second Hamiltonian structure, and which has a nontrivial Lax representation.
Abstract: It is shown that among a one‐parameter family of supersymmetric extensions of the Korteweg–de Vries equation, there is a special system that has an infinite number of conservation laws, which can be formulated in the second Hamiltonian structure, and which has a nontrivial Lax representation. Its modified version is also discussed.

344 citations


Journal ArticleDOI
TL;DR: In this article, the twisted product of functions on R2N is extended to a *-algebra of tempered distributions that contains the rapidly decreasing smooth functions, the distributions of compact support, and all polynomials, and moreover is invariant under the Fourier transformation.
Abstract: The twisted product of functions on R2N is extended to a *‐algebra of tempered distributions that contains the rapidly decreasing smooth functions, the distributions of compact support, and all polynomials, and moreover is invariant under the Fourier transformation The regularity properties of the twisted product are investigated A matrix presentation of the twisted product is given, with respect to an appropriate orthonormal basis, which is used to construct a family of Banach algebras under this product

311 citations


Journal ArticleDOI
TL;DR: In this article, the authors introduced new classes of symmetries for partial differential equations, which are neither point nor Lie-Backlund symmetric, and they are determined by a completely algorithmic procedure.
Abstract: New classes of symmetries for partial differential equations are introduced. By writing a given partial differential equation S in a conserved form, a related system T with potentials as additional dependent variables is obtained. The Lie group of point transformations admitted by T induces a symmetry group of S. New symmetries may be obtained for S that are neither point nor Lie–Backlund symmetries. They are determined by a completely algorithmic procedure. Significant new symmetries are found for the wave equation with a variable wave speed and the nonlinear diffusion equation.

303 citations


Journal ArticleDOI
TL;DR: In this article, a modified SU(2) chiral model in 2+1 dimensions is presented, in which the solitons move at constant velocity, and pass through one another without scattering or changing shape.
Abstract: There is a modified SU(2) chiral model in 2+1 dimensions which is integrable. It admits multisoliton solutions, in which the solitons move at constant velocity, and pass through one another without scattering or changing shape.

139 citations


Journal ArticleDOI
TL;DR: In this paper, the Moyal *-algebra is defined as an operator algebra over the space of smooth declining functions either on the configuration space or on the phase space itself.
Abstract: The topology of the Moyal *‐algebra may be defined in three ways: the algebra may be regarded as an operator algebra over the space of smooth declining functions either on the configuration space or on the phase space itself; or one may construct the *‐algebra via a filtration of Hilbert spaces (or other Banach spaces) of distributions. The equivalence of the three topologies thereby obtained is proved. As a consequence, by filtrating the space of tempered distributions by Banach subspaces, new sufficient conditions are given for a phase‐space function to correspond to a trace‐class operator via the Weyl correspondence rule.

129 citations


Journal ArticleDOI
TL;DR: In this article, the bi-Hamiltonian structure for a large class of hyberbolic systems of conservation laws in two field variables, including the equations of gas dynamics, shallow water waves, one-dimensional elastic media, and the Born-Infeld equation from nonlinear electrodynamics, is exhibited.
Abstract: The bi‐Hamiltonian structure for a large class of one‐dimensional hyberbolic systems of conservation laws in two field variables, including the equations of gas dynamics, shallow water waves, one‐dimensional elastic media, and the Born–Infeld equation from nonlinear electrodynamics, is exhibited. For polytropic gas dynamics, these results lead to a quadri‐Hamiltonian structure. New higher‐order entropy‐flux pairs (conservation laws) and higher‐order symmetries are exhibited.

126 citations


Journal ArticleDOI
TL;DR: In this paper, a generalization of the 2×2 Pauli matrices for the 3×3 case is described, and the remarkable properties of that basis are the grading of the Lie algebra it offers (each grading subspace is one dimensional) and the matrix group it generates [it is a finite group with the center of SL(n,C) as its commutator group].
Abstract: Properties of the Lie algebra gl(n,C) are described for a basis which is a generalization of the 2×2 Pauli matrices. The 3×3 case is described in detail. The remarkable properties of that basis are the grading of the Lie algebra it offers (each grading subspace is one dimensional) and the matrix group it generates [it is a finite group with the center of SL(n,C) as its commutator group].

114 citations


Journal ArticleDOI
TL;DR: In this paper, it is shown how a mathematical theory of generalized functions, in which the multiplications of distributions appearing in nonlinear equations of physics make sense, gives new formulas and new numerical results.
Abstract: It is shown how a mathematical theory of generalized functions, in which the multiplications of distributions appearing in nonlinear equations of physics make sense, gives new formulas and new numerical results. The new methods shown here are quite general but since each particular problem requires its own study, this paper is limited to elasticity and hydrodynamics. In elasticity Hooke’s law gives systems in a nonconservative form; the study of shock waves for these systems gives nonclassical multiplications of distributions of the form Y⋅δ (Y=Heaviside function, δ=Dirac mass at the origin). Using this new mathematical tool new formulas are obtained (more generally new numerical schemes): in a first step ‘‘ambiguous’’ results are obtained; then the ambiguity is removed. In hydrodynamics a formulation is obtained that has a nonconservative form and is at the basis of efficient new numerical schemes. Strictly speaking the reader is not assumed to know anything either on distributions or on elasticity and hydrodynamics, since the basic equations are recalled. All computations done in this paper are rigorous from the mathematical viewpoint.

109 citations


Journal ArticleDOI
TL;DR: The Lie algebra of the group of point transformations, leaving the Davey-Stewartson equations invariant, is obtained in this paper, and the general element of this algebra depends on four arbitrary functions of time.
Abstract: The Lie algebra of the group of point transformations, leaving the Davey–Stewartson equations (DSE’s) invariant, is obtained. The general element of this algebra depends on four arbitrary functions of time. The algebra is shown to have a loop structure, a property shared by the symmetry algebras of all known (2+1)‐dimensional integrable nonlinear equations. Subalgebras of the symmetry algebra are classified and used to reduce the DSE’s to various equations involving only two independent variables.

108 citations


Journal ArticleDOI
TL;DR: In this paper, the authors used renewal theory to analyze linear particle transport without scattering in a random mixture of two immiscible fluids, with the statistics described by arbitrary (non-Markovian) fluid chord length distributions.
Abstract: Renewal theory is used to analyze linear particle transport without scattering in a random mixture of two immiscible fluids, with the statistics described by arbitrary (non‐Markovian) fluid chord length distributions. One conclusion (for unimodal distributions) that is drawn is that the mean and variance of the chord length distributions through each fluid is sufficient knowledge of the statistics to give a reasonably accurate description of the ensemble averaged intensity. Expressions for effective cross sections and an effective source to be used in the usual deterministic transport equation are also obtained. The use of these effective quantities allows statistical information to be introduced very simply into a standard transport equation. An analysis is given which shows how the transport description, including scattering, in a Markovian mixture can be modified to yield an approximate description of transport in a non‐Markovian mixture. Numerical results are given to assess the accuracy of this model, as well as the simpler model involving the effective cross sections and source.

94 citations


Journal ArticleDOI
TL;DR: In this article, all real Lie algebras of dimension up to 8 that admit a nontrivial Levi decomposition was found, and all of them admit a non-parametric nonparametric decomposition.
Abstract: All real Lie algebras of dimension up to 8 that admit a nontrivial Levi decomposition are found.

Journal ArticleDOI
TL;DR: In this paper, a method to derive conservation laws for evolution equations that describe pseudospherical surfaces is introduced based on a geometrical property of these surfaces and a new third-order evolution equation is obtained as a first example for a nongeneric case in the classification given by Chern and Tenenblat.
Abstract: A method to derive conservation laws for evolution equations that describe pseudospherical surfaces is introduced based on a geometrical property of these surfaces. A new third‐order evolution equation is obtained as a first example for a nongeneric case in the classification given by Chern and Tenenblat [Stud. Appl. Math. 74, 1 (1986)].

Journal ArticleDOI
TL;DR: In this article, the Cauchy problem and the propagation of discontinuities for stringy gravity with Gauss-Bonnet terms were studied in arbitrary dimensions, and the authors showed that the problem is NP-hard.
Abstract: The Cauchy problem and the propagation of discontinuities for stringy gravity—that is, equations with Gauss–Bonnet terms—in arbitrary dimensions are studied.

Journal ArticleDOI
TL;DR: In particular, simple rules for determining the SO(2n)×Sp(2m,R) decomposition of the unitary lowest weight representations of OSp (2n/2m) with even subgroups were derived in this paper.
Abstract: The oscillator construction of the unitary irreducible lowest (highest) weight representations of the noncompact supergroup OSp(2n/2m,R) with even subgroup SO(2n)×Sp(2m,R) is given. In particular, simple rules for determining the SO(2n)×Sp(2m,R) decomposition of the unitary lowest weight representations of OSp(2n/2m,R) are derived.

Journal ArticleDOI
TL;DR: In this paper, a necessary and sufficient condition is given for the symmetry algebra of a general linear system of second-order ODEs to be of maximal dimension (i.e., n2+4n+3) and isomorphic to sl(n+2,R), the well-known symmetry algebra for the free-particle equation x = 0.
Abstract: In this paper, several problems concerning the Lie algebra structure of symmetries and variational symmetries of a general linear system of second‐order ordinary differential equations are studied. In particular, a necessary and sufficient condition is obtained, in terms of the coefficients of the system, for the system’s symmetry algebra to be of maximal dimension (i.e., n2+4n+3) and isomorphic to sl(n+2,R), the well‐known symmetry algebra of the free‐particle equation x‘=0. When this condition is satisfied, it is proved that the system is Lagrangian and that its variational symmetry algebra is isomorphic to a fixed, (n2+3n+6)/2‐ dimensional Lie algebra, whose structure (Levi–Mal’cev decomposition and realization by means of a matrix algebra) is determined. For the particular case of isotropic systems (which includes, as far as is known, all the examples treated in the literature), explicit formulas for the generators of both the symmetry algebra and the variational symmetry algebra are obtained.

Journal ArticleDOI
TL;DR: The solitary wave interaction mechanism for the good Boussinesq equation was investigated and found to be far more complicated than was previously thought as discussed by the authors, and the existence of a potential well was linked to the different behaviors observed between small and large initial conditions.
Abstract: The solitary‐wave interaction mechanism for the good Boussinesq equation is investigated and found to be far more complicated than was previously thought. Three salient features are that solitary waves only exist for a finite range of velocities, that large solitons can turn into so‐called antisolitons, and that it is possible for solitons to merge and split. Small solitons, however, appear to be stable. The existence of a potential well is linked to the different behaviors observed between small and large initial conditions.

Journal ArticleDOI
TL;DR: In this article, the group theoretical concepts of embedded representations and dynamical structure groups are introduced in order to describe the common physical situation in which collective bands of states of a manybody system are well described by an algebraic collective model even though the states may not span an invariant subspace of the manybody Hilbert space.
Abstract: The group theoretical concepts of embedded representations and dynamical structure groups, distinct from dynamical symmetry groups, are introduced in order to describe the common physical situation in which collective bands of states of a many‐body system are well described by an algebraic collective model even though the states may not span an invariant subspace of the many‐body Hilbert space.

Journal ArticleDOI
TL;DR: In this paper, the vector coherent state inner product can be inferred algebraically, by K-matrix theory, and changed to a simpler Bargmann inner product thereby facilitating the explicit calculation of the matrix representaions of Lie algebras.
Abstract: Vector coherent state theory is developed and presented in a form that explicitly exhibits its general applicability to the ladder representations of all semisimple Lie groups and their Lie algebras. It is shown that, in a suitable basis, the vector coherent state inner product can be inferred algebraically, by K‐matrix theory, and changed to a simpler Bargmann inner product thereby facilitating the explicit calculation of the matrix representaions of Lie algebras. Applications are made to the even and odd orthogonal Lie algebras.

Journal ArticleDOI
TL;DR: In this article, the authors studied two-dimensional and three-dimensional billiard systems with elliptical and ellipsoidal boundaries, respectively, and established generalized Poncelet's theorem in three dimensions.
Abstract: Two‐ and three‐dimensional billiard systems with elliptical and ellipsoidal boundaries, respectively, are studied. It is known that the trajectory of such a two‐dimensional system generates a caustic conic curve. Many properties of the elliptical billiard system in the language of projective geometry can be described. One of these properties is that if a trajectory is closed after p bounces, then all trajectories sharing the same caustic conic are also closed after p bounces. In projective geometry, this is known as Poncelet’s theorem. Many of the two‐dimensional results are extended to three dimensions. In particular, it is shown that all straight segments of a trajectory in a three‐dimensional ellipsoidal billiard system are always tangent to two caustic quadrics. If a trajectory is closed after p bounces, then all trajectories sharing the same two caustic quadrics are also closed after p bounces. A generalized Poncelet’s theorem in three dimensions is thus established. On the basis of numerical studies, it is conjectured that this generalized Poncelet’s theorem is also valid for an arbitrary finite number field. The implications of the results are discussed and their extension to n dimensions is outlined.

Journal ArticleDOI
TL;DR: In this paper, the solutions of the equation y+yy+βy3 = 0, where β is a free parameter, are investigated and the analytical asymptotic solutions and their behavior are given according to the value of β and to the initial conditions.
Abstract: The solutions of the equation y+yy+βy3=0, where β is a free parameter, are investigated. For β= (1)/(9) the equation is linearizable through an eight‐parameter symmetry group and is completely integrable. For β≠ (1)/(9) only two symmetries subsist, but through a dynamical description the analytical asymptotic solutions and their behavior are given according to the value of β and according to the initial conditions.

Journal ArticleDOI
TL;DR: In this article, a Poincare-invariant scalar product and corresponding physical Hilbert space of states are constructed by finding a tensor current of rank 2, jμν(x1,x2), satisfying two independent conservation laws, relative to particles 1 and 2, respectively.
Abstract: In the framework of two‐particle relativistic quantum mechanics, a Poincare‐invariant scalar product and the corresponding physical Hilbert space of states are constructed. This is achieved by finding a tensor current of rank 2, jμν(x1,x2), satisfying two independent conservation laws, relative to particles 1 and 2, respectively. Then the scalar product is obtained by integrating the current jμν over two three‐dimensional spacelike hypersurfaces. The Hermiticity of the Poincare group generators is ensured by the fact that the kernel of the current jμν is translation invariant and covariant. A simple expression of the scalar product is obtained when one chooses for the two spacelike hypersurfaces two constant parallel hyperplanes. The positivity of the norm is, in general, ensured if the spectrum of the eigenvalues of the total mass squared operator comes out to be positive.

Journal ArticleDOI
TL;DR: In this article, the existence of affine collineations in space-time is discussed and the types of space time admitting proper affine co-lineations are discussed.
Abstract: The existence of affine collineations in space‐time is discussed and the types of space‐time admitting proper affine collineations is displayed. The close connection between such space‐times and their holonomy structure and local decomposability is established. Affine collineations with fixed points are also considered as is the problem of extending local affine collineations to the whole of space‐time.

Journal ArticleDOI
TL;DR: In this article, the generalized Bogoliubov transformation for mixed systems of bosons and fermions is also obtained for the simple harmonic oscillator, and the invariant integration measure is calculated by studying transformation properties of supercoset variables.
Abstract: Coherent states for the harmonic oscillator representations of the noncompact supergroup Osp(1/2N,R) are introduced and the invariant integration measure is calculated by studying transformation properties of supercoset variables. The generalized Bogoliubov transformation for mixed systems of bosons and fermions is also obtained. An example for the simple harmonic oscillator is given.

Journal ArticleDOI
TL;DR: In this paper, two approaches to construct the strong coupling expansion for the anharmonic oscillator with the potential V(x)= 1/2 x2+g/4)x4 are proposed.
Abstract: Two novel approaches to construction of the strong coupling expansion for the anharmonic oscillator with the potential V(x)= 1/2 x2+(g/4)x4 are proposed. The first one is simply a straightforward solution of the Schrodinger equation via the ‘‘nonlinearization’’ technique, resulting in the rapidly convergent perturbation series. The second one is a version of the path integral perturbation theory, but with an unconventional choice of the zeroth approximation action. Nine leading coefficients of the strong coupling expansion are computed. They decrease rapidly, the ninth one being of the order of 10−9. Three leading corrections of the nonlinearization approach provide the ground‐state energy within a relative accuracy of 10−7–10−9, at an arbitrary coupling g. The explicit formulas for corrections enable one to study the analyticity properties of the energy as a function of the coupling g. In the second approach the strong coupling expansion coefficients are computed as an infinite linear series of the weak ...

Journal ArticleDOI
TL;DR: In this paper, the Hamiltonian system given by H = 1/2 ǫp2+V(q) with V∈C∞(Rn) is considered.
Abstract: The Hamiltonian system given by H= 1/2 p2+V(q) with V∈C∞(Rn) is considered. A method for integrating such a system is that of separating the variables in the Hamilton–Jacobi equation. It is known that if such a separation is possible, then it can take place only when the equation is expressed in terms of generalized elliptic coordinates or in a degeneration of these. A criterion is proposed for deciding if separation is possible, and if it is, in which degeneration of elliptic coordinates it takes place.

Journal ArticleDOI
TL;DR: In this article, a noncanonical Poisson bracket for the Hamiltonian dynamics of an ideal spin glass is shown to be identical to that for the dynamics of a Yang-Mills fluid plasma, although the Hamiltonians differ for the two theories.
Abstract: A dictionary of correspondence is established between the dynamical variables for spin‐glass fluid and Yang‐Mills plasma. The Lie‐algebraic interpretation of these variables is presented for the two theories. The noncanonical Poisson bracket for the Hamiltonian dynamics of an ideal spin glass is shown to be identical to that for the dynamics of a Yang–Mills fluid plasma, although the Hamiltonians differ for the two theories. This Poisson bracket is associated to the dual space of an infinite‐dimensional Lie algebra of semidirect‐product type.

Journal ArticleDOI
TL;DR: In this article, the exact formulas that describe the probability that such a system may be found in a pure state with a given symmetry with respect to permutations of atoms are presented and the asymptotic form of these probabilities valid for large n is also derived.
Abstract: Suppose that the state of a system of N n‐level atoms is given by a tensor product of N identical density matrices. The exact formulas are presented that describe the probability that such a system may be found in a pure state with a given symmetry with respect to permutations of atoms. The asymptotic form of these probabilities valid for large N is also derived.

Journal ArticleDOI
TL;DR: The connection between τ functions and zero curvature equations for the homogeneous construction of the basic module L(Λ0) over the simplest affine Kac-Moody algebra A(1)1 is studied in this paper.
Abstract: The connection between τ functions and zero curvature equations for the homogeneous construction of the basic module L(Λ0) over the simplest affine Kac–Moody algebra A(1)1 is studied.

Journal ArticleDOI
TL;DR: In this paper, two types of singular boundaries arising in these solutions are examined by verifying the local behavior of causal curves approaching these boundaries, and a criterion due to C. S. Clarke is given, allowing one to test the completeness of arbitrary accelerated timelike curves in terms of their acceleration and proper time.
Abstract: Geometrical and physical properties of the solutions derived and classified in Part I [J. Math. Phys. 28, 1118 (1987)] are examined in detail. It is shown how the imposition of zero shear restricts the possible choices of equations of state. Two types of singular boundaries arising in these solutions are examined by verifying the local behavior of causal curves approaching these boundaries. For this purpose, a criterion due to C. J. S. Clarke (private communication) is given, allowing one to test the completeness of arbitrary accelerated timelike curves in terms of their acceleration and proper time. One of these boundaries is a spacelike singularity at which causal curves terminate as pressure diverges but matter‐energy and charge densities remain finite. At the other boundary, which is timelike if the expansion Θ is finite, proper volume of local fluid elements vanishes as all state variables diverge but causal curves are complete. If Θ diverges at this boundary, a null singularity arises as the end pro...

Journal ArticleDOI
TL;DR: In this paper, the oscillator method for the construction of unitary highest weight representations of noncompact groups and supergroups is generalized, and the generalized supercoherent states associated with these unitary representations are also defined.
Abstract: The oscillator method for the construction of unitary highest (or lowest) weight representations of noncompact groups and supergroups is generalized. Within this generalization, the method yields unitary highest weight representations of all simple supergroups whose even subgroups are in the form of a direct product of a compact group with a simple noncompact group. The method is illustrated by studying in detail the unitary highest weight representations of the supergroup OSp (2n+1/2m,R). The generalized supercoherent states associated with these unitary representations are also defined.