Showing papers in "Journal of Mathematical Physics in 1983"
TL;DR: In this paper, the authors define the Painleve property for partial differential equations and show how it determines, in a remarkably simple manner, the integrability, the Backlund transforms, the linearizing transforms, and the Lax pairs of three well-known partial differential equation (Burgers' equation, KdV equation, and modified KDV equation).
Abstract: In this paper we define the Painleve property for partial differential equations and show how it determines, in a remarkably simple manner, the integrability, the Backlund transforms, the linearizing transforms, and the Lax pairs of three well‐known partial differential equations (Burgers’ equation, KdV equation, and the modified KdV equation). This indicates that the Painleve property may provide a unified description of integrable behavior in dynamical systems (ordinary and partial differential equations), while, at the same time, providing an efficient method for determining the integrability of particular systems.
1,958 citations
TL;DR: In this article, the authors investigated the Painleve property for partial differential equations and showed that it is invariant under the Moebius group (acting on dependent variables) and obtained the appropriate Lax pair for the underlying nonlinear pde.
Abstract: In this paper we investigate the Painleve property for partial differential equations. By application to several well‐known partial differential equations (Burgers, KdV, MKdV, Bousinesq, higher‐order KdV and KP equations) it is shown that consideration of the ‘‘singular manifold’’ leads to a formulation of these equations in terms of the ‘‘Schwarzian derivative.’’ This formulation is invariant under the Moebius group (acting on dependent variables) and is shown to obtain the appropriate Lax pair (linearization) for the underlying nonlinear pde.
572 citations
TL;DR: In this paper, the Schrodinger equation for a quantum object influenced by adjustable external fields provides a state-evolution equation which is linear in ψ and linear in the external controls (thus a bilinear control system).
Abstract: The systems‐theoretic concept of controllability is elaborated for quantum‐mechanical systems, sufficient conditions being sought under which the state vector ψ can be guided in time to a chosen point in the Hilbert space H of the system. The Schrodinger equation for a quantum object influenced by adjustable external fields provides a state‐evolution equation which is linear in ψ and linear in the external controls (thus a bilinear control system). For such systems the existence of a dense analytic domain Dω in the sense of Nelson, together with the assumption that the Lie algebra associated with the system dynamics gives rise to a tangent space of constant finite dimension, permits the adaptation of the geometric approach developed for finite‐dimensional bilinear and nonlinear control systems. Conditions are derived for global controllability on the intersection of Dω with a suitably defined finite‐dimensional submanifold of the unit sphere SH in H. Several soluble examples are presented to illuminate th...
379 citations
TL;DR: In this article, the exterior Cauchy problem for the fourth-order theories of gravity derived from the Lagrangian densities L=(−g) 1/2 (R+ (1/2)aR2+bRμν Rμν) −κLm was discussed.
Abstract: The exterior Cauchy problem is discussed for the fourth‐order theories of gravity derived from the Lagrangian densities L=(−g)1/2 (R+ (1/2)aR2+bRμν Rμν) −κLm. When b≠0, the Cauchy problem can be solved by the standard method already used in general relativity. When b=0, the problem cannot be formulated as in the case where b≠0, since the corresponding fourth‐order theory is shown to be equivalent to a second‐order scalar–tensor theory. This scalar–tensor theory is proved to coincide with one of the models of gravity proposed by O’Hanlon in order to present a covariant version of the massive dilaton theory suggested by Fujii. This result is generalized: The models of O’Hanlon are shown to be indistinguishable from the fourth‐order theories derived from the Lagrangian densities L=(−g)1/2 F(R)−κLm, where F is any real function such that F″(R) does not identically vanish.
274 citations
TL;DR: In this paper, the evolution of envelope solitons in the presence of perturbations is discussed in terms of the functional behavior of N(η2), where η2 is the nonlinear frequency shift.
Abstract: Envelope soliton solutions of a class of generalized nonlinear Schrodinger equations are investigated. If the quasiparticle number N is conserved, the evolution of solitons in the presence of perturbations can be discussed in terms of the functional behavior of N(η2), where η2 is the nonlinear frequency shift. For ∂η2N >0, the system is stable in the sense of Liapunov, whereas, in the opposite region, instability occurs. The theorem is applied to various types of envelope solitons such as spikons, relatons, and others.
247 citations
TL;DR: In this paper, the results of a systematic investigation of invariance properties of a large class of nonlinear evolution equations under a one-parameter continuous (Lie) group of transformations are presented.
Abstract: We present the results of a systematic investigation of invariance properties of a large class of nonlinear evolution equations under a one‐parameter continuous (Lie) group of transformations. It is shown that, in general, the corresponding invariant variables (the subclass of which is the usual similarity variables) lead to ordinary differential equations of Painleve type in the case of inverse scattering transform solvable equations, as conjectured by Ablowitz, Ramani, and Segur. This is found to be also true for certain higher spatial dimensional versions such as the Kadomtsev–Petviashivilli, two dimensional sine–Gordon, and Ernst equations. For the nonsolvable equations considered here this invariance study leads to ordinary differential equations with movable critical points.
134 citations
TL;DR: In this paper, the Dirac-Bergmann generalized Hamiltonian dynamics for a degenerate Lagrangian system is formulated on the Whitney sum T*Q⊕TQ of the phase space T *Q and the velocity space TQ over the configuration space Q.
Abstract: The Dirac–Bergmann generalized Hamiltonian dynamics for a degenerate‐Lagrangian system is formulated on the Whitney sum T*Q⊕TQ of the phase space T*Q and the velocity space TQ over the configuration space Q. The formulation is related to those on T*Q and TQ. Some ambiguities concerning generalized dynamics that have appeared in the literature are clarified.
131 citations
TL;DR: In this paper, it was shown that for systems with an arbitrary number of degrees of freedom, a necessary and sufficient condition for the Wigner function to be nonnegative is that the corresponding state wave function is the exponential of a quadratic form.
Abstract: It is shown that, for systems with an arbitrary number of degrees of freedom, a necessary and sufficient condition for the Wigner function to be nonnegative is that the corresponding state wavefunction is the exponential of a quadratic form. This result generalizes the one obtained by Hudson [Rep. Math. Phys. 6, 249 (1974)] for one‐dimensional systems.
122 citations
TL;DR: In this paper, a procedure for the computation of nonlinear effects of arbitrarily high degree, and explicit formulas are given through effects of degree 5, is presented for trajectories near a given trajectory for general Hamiltonian systems.
Abstract: Lie algebraic methods are developed to describe the behavior of trajectories near a given trajectory for general Hamiltonian systems. A procedure is presented for the computation of nonlinear effects of arbitrarily high degree, and explicit formulas are given through effects of degree 5. Expected applications include accelerator design, charged particle beam and light optics, other problems in the general area of nonlinear dynamics, and, perhaps, with suitable modification, the area of S‐matrix expansions in quantum field theory.
118 citations
TL;DR: In this paper, the weak painleve property is applied to the case of simple Hamiltonians describing the motion of a particle in two-dimensional polynomial potentials of degree three and four.
Abstract: The weak‐Painleve property, as a criterion of integrability, is applied to the case of simple Hamiltonians describing the motion of a particle in two‐dimensional polynomial potentials of degree three and four. This allows a complete identification of all the integrable cases of cubic potentials. In the case of quartic potentials, although our results are not exhaustive, some new integrable cases are discovered. In both cases the integrability is explicited by a direct calculation of the second integral of motion of the system.
111 citations
TL;DR: In this paper, a family of dynamical systems associated with the motion of a particle in two space dimensions is presented, which are completely integrable and have a second integral of motion quadratic in velocities.
Abstract: We present a family of dynamical systems associated with the motion of a particle in two space dimensions. These systems possess a second integral of motion quadratic in velocities (apart from the Hamiltonian) and are thus completely integrable. They were found through the derivation and subsequent resolution of the integrability condition in the form of a partial differential equation (PDE) for the potential. A most important point is that the same PDE was derived through considerations on the analytic structure of the singularities of the solutions (‘‘weak‐Painleve property’’).
TL;DR: In this article, a superposition rule for the matrix Riccati equation (MRE) W=A+WB+CW+WDW was derived, expressing the general solution in terms of five known solutions for all n ≥ 2.
Abstract: A superposition rule is obtained for the matrix Riccati equation (MRE) W=A+WB+CW+WDW [where W(t), A(t), B(t), C(t), and D(t) are real n×n matrices], expressing the general solution in terms of five known solutions for all n≥2. The symplectic MRE (W=WT, A=AT, D=DT, C=BT) is treated separately, and a superposition rule is derived involving only four known solutions. For the ‘‘unitary’’ and GL(n,R) subcases (with D=A and C=BT, or D=−A and C=BT, respectively), superposition rules are obtained involving only two solutions. The derivation of these results is based upon an interpretation of the MRE in terms of the action of the groups SL(2n,R), SP(2n,R), U(n), and GL(n,R) on the Grassman manifold Gn(R2n).
TL;DR: In this article, a definition of multipole moments for stationary asymptotically flat solutions of Einstein's equations is proposed, and it is shown that these moments characterize a given space-time uniquely.
Abstract: A definition of multipole moments for stationary asymptotically flat solutions of Einstein’s equations is proposed. It is shown that these moments characterize a given space‐time uniquely. Conversely, they can be arbitrarily prescribed, i.e., they generate power series for the field variables which satisfy the field equations to all orders. Despite their apparently rather different origin, they are shown to be identical with the Geroch–Hansen ones.
TL;DR: The difference between classical mechanics and quantum mechanics can be found in the nature of the observables that are considered for the physical system under consideration as discussed by the authors, and the difference can be explained by the fact that classical mechanics can only describe a certain kind of what we called ‘classical observable; it cannot describe, however, classical observables.
Abstract: We analyze the difference between classical mechanics and quantum mechanics. We come to the conclusion that this difference can be found in the nature of the observables that are considered for the physical system under consideration. Classical mechanics can only describe a certain kind of what we called ‘‘classical observable.’’ Quantum mechanics can only describe another kind of observable; it cannot describe, however, classical observables. To perform this analysis, we use a theory where every kind of observable can be treated and which is in a natural way a generalization of both classical and quantum mechanics. If in a study of a physical system in this theory we restrict ourselves to the classical observables, we rediscover classical mechanics as a kind of first study of the physical system, where all the nonclassical properties are hidden. If we find that this first study is too rough we can also study the nonclassical part of the physical system by a theory which is eventually quantum mechanics.
TL;DR: In this paper, the classical complex and real Lie and Jordan algebras with involutions are classified into conjugacy classes under the action of the corresponding classical Lie group.
Abstract: Elements of the classical complex and real Lie and Jordan algebras with involutions are classified into conjugacy classes under the action of the corresponding classical Lie group. Normal forms of representatives of each conjugacy class are chosen so as to resemble the Jordan normal forms of n×n complex matrices. For completeness similar results are given for gl(n,C), gl(n,R), and gl(n,H).
TL;DR: In this paper, the relation between Kac-Dynkin diagrams and supertableaux is discussed, and the relationship between super-tableaux and super-kac diagrams is discussed.
Abstract: We show the relation between Kac–Dynkin diagrams and supertableaux. We show the relation between Kac–Dynkin diagrams and supertableaux.
TL;DR: In this paper, a new nonlinear field theory based on three principles taken from empiricism was established, which allows us to describe frictional effects in dissipative systems with the aid of a Schrodinger-type field equation with logarithmic nonlinearity.
Abstract: Based on three principles taken from empiricism we establish a new nonlinear field theory which allows us to describe frictional effects in dissipative systems with the aid of a Schrodinger‐type field equation with logarithmic nonlinearity. This nonlinear field equation corresponds to the classical Langevin equation and can be interpreted in different ways, taking the view of classical undulatory theory or probabilistic theories, respectively. Because of formal similarities, calculations can be performed independently of the accepted interpretation; only the occurring quantities have to be provided with the corresponding meaning. As an example, the nonlinear field equation for the damped harmonic oscillator is solved exactly. The solutions, a wavefunction and a wavepacketlike solution, a solution with Gaussian shape, exhibit reasonable properties, contain the correct reduced frequency Ω=(ω20−γ2/4)1/2 and are different from the solutions of the undamped problem. The properties of our nonlinear friction ter...
TL;DR: In this paper, a null cone formulation of axially symmetric gravitational and matter fields is used for the production of gravitational waves, both analytically and numerically, using a single conformally compactified patch, which is well suited for numerical computation.
Abstract: The production of gravitational waves is explored, both analytically and numerically, using a null cone formulation of axially symmetric gravitational and matter fields. The coupled field equations are written in an integral form, on a single conformally compactified patch, which is well suited for numerical computation. Some analytic and numerical solutions of the initial value problem are given. The total mass and radiation flux is studied in detail for a special class of collapsing dust configurations.
TL;DR: Several notions of invariance and covariance for products with respect to Lie algebras and Lie groups are investigated in this article, including the Poincare group and the Lie group covariance.
Abstract: Several notions of invariance and covariance for * products with respect to Lie algebras and Lie groups are investigated. Some examples, including the Poincare group, are given. The passage from the Lie‐algebra invariance to the Lie‐group covariance is performed. The compact and nilpotent cases are treated.
TL;DR: The explicit construction of a dense subspace Φ of square integrable functions on the positive half of the real line is given in this paper, which has the properties that it is endowed with a metrizable nuclear topology, it is stable under multiplication by x, and the functions in Φ have suitable analytical continuation to a half plane.
Abstract: The explicit construction of a dense subspace Φ of square integrable functions on the positive half of the real line is given. This space Φ has the properties that: (1) it is endowed with a metrizable nuclear topology, (2) it is stable under multiplication by x, and (3) the functions in Φ have suitable analytical continuation to a half plane. The space Φ* of functions which are conjugate to elements of Φ is also considered. Then the triplets Φ⊆ L2 (0,∞)⊆Φ′ and Φ*⊆ L2 (0,∞)⊆Φ*′ are used to give a description of resonances.
TL;DR: The generalized Langevin equation was first derived by Mori using the Gram-Schmidt orthogonalization process as discussed by the authors, which can also be derived by a method of recurrence relations.
Abstract: The generalized Langevin equation was first derived by Mori using the Gram‐Schmidt orthogonalization process. This equation can also be derived by a method of recurrence relations. For a physical space commonly used in statistical mechanics, the recurrence relations are simple and they lead directly to the Langevin equation. The Langevin equation is shown to be composed of one homogeneous and one inhomogeneous equation.
TL;DR: In this paper, a graded tensor calculus corresponding to arbitrary abelian groups of degrees and arbitrary commutation factors is developed, and the standard basic constructions and definitions, like tensor products, spaces of multilinear mappings, contractions, symmetrization, symmetric algebra, as well as the transpose, adjoint, and trace of a linear mapping, are generalized to the graded case and a multitude of canonical isomorphisms is presented.
Abstract: We develop a graded tensor calculus corresponding to arbitrary abelian groups of degrees and arbitrary commutation factors. The standard basic constructions and definitions, like tensor products, spaces of multilinear mappings, contractions, symmetrization, symmetric algebra, as well as the transpose, adjoint, and trace of a linear mapping, are generalized to the graded case and a multitude of canonical isomorphisms is presented. Moreover, the graded versions of the classical Lie algebras are introduced, and some of their basic properties are described.
TL;DR: In this paper, the occupation numbers of different levels of the imperfect Bose gas are computed and the fluctuations of the overall density are on the contrary normal, Gaussian, and shape independent.
Abstract: The fluctuations of the occupation numbers of the different levels of the imperfect Bose gas are computed. They are neither normal, Gaussian, nor independent of the way the infinite volume limit is taken. The fluctuations of the overall density are on the contrary normal, Gaussian, and shape independent.
TL;DR: In this paper, the general form of dynamics which preserve the set of closed linear submanifolds (i.e., properties) is deduced, and the result generalizes Wigner's theorem and provides a model of some irreversible phenomena like spin relaxation, damped oscillator, etc.
Abstract: General dynamics compatible with the Hilbert space structure of quantum kinematics are considered. The general form of dynamics which preserve the set of closed linear submanifolds (i.e., properties) is deduced. Since the orthogonality relation is not necessarily preserved, the result generalizes Wigner’s theorem and provides a model of some irreversible phenomena like spin relaxation, damped oscillator, etc. Connections with quantum logic and with statistical mechanics are presented.
TL;DR: In this article, the authors derived bounds on the energy trajectories Enl =Fnl(v) of the Hamiltonian H =−Δ+vf(r), where v is a coupling constant.
Abstract: Suppose f(r) is an attractive central potential of the form f(r)=∑ki=1 g(i)( f(i)(r)), where {f(i)} is a set of basis potentials (powers, log, Hulthen, sech2) and {g(i)} is a set of smooth increasing transformations which, for a given f, are either all convex or all concave. Formulas are derived for bounds on the energy trajectories Enl =Fnl(v) of the Hamiltonian H=−Δ+vf(r), where v is a coupling constant. The transform Λ( f)=F is carried out in two steps: f→f→F, where f(s) is called the kinetic potential of f and is defined by f(s)=inf(ψ,f,ψ) subject to ψ∈D⊆L2(R3), where D is the domain of H, ∥ψ∥=1, and (ψ,−Δψ)=s. A table is presented of the basis kinetic potentials { f(i)(s)}; the general trajectory bounds F*(v) are then shown to be given by a Legendre transformation of the form (s, f*(s)) →(v, F*(v)), where f*(s) =∑ki=1g(i)× ( f(i)(s)) and F*(v) =mins>0{s+v f*(s)}. With the aid of this potential construction set (a kind of Schrodinger Lego), ground‐state trajectory bounds are derived for a vari...
TL;DR: The geometrical structure for a theory of gravitation, based on a nonsymmetric metric in a four-dimensional real manifold, was developed in this article, where the local fiber bundle gauge group is GL(4,R), which contains the (local) homogeneous Lorentz gauge group SO(3,1) of general relativity.
Abstract: The geometrical structure is developed for a theory of gravitation, based on a nonsymmetric metric in a four‐dimensional real manifold. The local fiber bundle gauge group is GL(4,R), which contains the (local) homogeneous Lorentz gauge group SO(3,1) of general relativity.
TL;DR: Inequalities for Fourier transforms are developed in this article which describe local uncertainty principles in the sense that if the uncertainty of momentum is small, then so is the probability of being localized at any point.
Abstract: Inequalities for Fourier transforms are developed which describe local uncertainty principles in the sense that if the uncertainty of momentum is small, then so is the probability of being localized at any point. They give estimates for essentially all states and lead to lower bounds for Hamiltonians.
TL;DR: In this paper, an analytic-numeric real axis integration technique has been developed for such integrals and it is combined with piecewise sinusoidal expansions to solve the Fredholm integral equation for the unknown current density.
Abstract: Printed circuit antennas are becoming an integral part of imaging arrays in microwave, millimeter, and submillimeter wave frequencies. The electrical characteristics of such antennas can be analyzed by solving integral equations of the Fredholm first kind. The kernel involves Sommerfeld integrals which are particularly difficult to solve when source and field points lie on an electrical discontinuity, as it occurs in the determination of the characteristics of printed circuit antennas. An analytic‐numeric real axis integration technique has been developed for such integrals and it is combined with piece‐wise sinusoidal expansions to solve the Fredholm integral equation for the unknown current density.
TL;DR: In this paper, the problem of the lamb shift and the spontaneous emission of light in a framework of nonrelativistic quantum electrodynamics by using an exactly soluble model of a harmonic oscillator atom interacting with a quantized electromagnetic field was studied.
Abstract: We study rigorously the problem of the lamb shift and the spontaneous emission of light in a framework of nonrelativistic quantum electrodynamics by using an exactly soluble model of a harmonic oscillator atom interacting with a quantized electromagnetic field. We show that, under the perturbation of the electromagnetic field, all the point spectra corresponding to the excited states of the unperturbed atom disappear. This means that the ‘‘energy level shifts’’ (Lamb shifts) of the excited states of the atom cannot be described simply in terms of shifts of point spectra. Then, we give a rigorous mathematical meaning to both formal perturbation theories for the ‘‘energy level shifts’’ and for the transitions of the excited states due to the spontaneous emission of light, showing that the ‘‘energy level shifts’’ and the ‘‘decay probabilities’’ of the excited states of the atom are characterized in terms of the resonance pole of the S‐matrix for the photon scattering by the atom. We also discuss broken symmetry aspects and infinite mass‐renormalization of the model.
TL;DR: In this paper, the authors considered the initial value problem for the conformally coupled scalar field and higher derivative gravity by expressing the equations of each theory in harmonic coordinates, and showed that the (vacuum) equations can take the form of a diagonal hyperbolic system with constraints on the initial data.
Abstract: The initial value problem is considered for the conformally coupled scalar field and higher derivative gravity, by expressing the equations of each theory in harmonic coordinates. For each theory it is shown that the (vacuum) equations can take the form of a diagonal hyperbolic system with constraints on the initial data. Consequently these theories possess well‐posed initial value formulations.