Journal of Mathematical Physics
Công bố khoa học tiêu biểu
* Dữ liệu chỉ mang tính chất tham khảo
Sắp xếp:
Separation of the Interaction Potential into Two Parts in Treating Many-Body Systems. I. General Theory and Applications to Simple Fluids with Short-Range and Long-Range ForcesSystematic methods are developed for investigating the correlation functions and thermodynamic properties of a classical system of particles interacting via a pair potential v(r) = q(r) + w(r). The method is then applied to the case in which w(r) is a ``Kac potential'' w(r, γ) = γvφ(γr) (v the dimensionality of the space) whose range γ−1 is very long compared to the range of q(r). Our work is related closely to the work of Kac, Uhlenbeck, and Hemmer. The main new feature of our method is the separation of the correlations, e. g., the two-particle Ursell function ℱ(r), into a short-range part ℱs(r, γ) and a long-range part ℱL(y, γ), y ≡ γ r; r the distance between the particles. The two parts of ℱ are defined in terms of their representation by graphs with density (or fugacity) vertices and K-and φ-bonds, K(r) = e−-βq − 1, Φ = −βw. A resummation of these graphs then yields a simple graphical representation for the long-range part of the correlation functions in terms of graphs with φ-bonds and ``hypervertices'' made up of the short-range part of the correlations. This representation is then used in this paper to make separate expansions of ℱs(r, γ) and ℱL(y, γ) and through them of the thermodynamic parameters in powers of γ. Explicit calculations of the Helmboltz free energy is carried out to a higher order in γ than done previously by Hemmer and it is shown how to carry out the calculation, in principle, to any order. The general method is further applied (in separate articles) to lattice gases, plasmas, and to the special problem of critical phenomena.
Journal of Mathematical Physics - Tập 6 Số 8 - Trang 1282-1298 - 1965
Soliton solutions to coupled higher-order nonlinear Schrödinger equationsA set of coupled higher-order nonlinear Schrödinger 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.
Journal of Mathematical Physics - Tập 33 Số 3 - Trang 1208-1215 - 1992
A restricted Bäcklund transformationThe Bäcklund transformation provides a mathematical tool which displays the interaction of solitons. Here a simple, systematic Bäcklund formalism is introduced which permits the explicit construction of these transformations for a restricted class of nonlinear wave equations. Traditionally a Bäcklund transformation has been viewed as a transformation of a solution surface of a partial differential equation into another surface which may not satisfy the same equation. In the present paper the term ``restricted Bäcklund transformation'' (hereafter abbreviated R-B) is used to refer to the case in which the transformed surface does satisfy the same equation. This formalism clarifies the nature of those transformations which have already been used to study nonlinear interactions in many physical problems. The formalism is introduced through a form of the linear Klein-Gordon equation. For this linear example a complete set of Fourier components is generated by a sequence of R-B transformations. This concrete example also indicates the type of results one can expect in the nonlinear case. For the nonlinear equation φx y = F(φ), a theorem is established which states that R-B transformations exist if and only if the nonlinearity F(·) satisfies F″ = κF, where κ is a constant. For such nonlinearities, the R-B transformations are explicitly constructed and are used to display exact nonlinear interactions. A relationship between the condition F″ = κF, the existence of an infinite number of conservation laws, and the transformation theory is briefly discussed.
Journal of Mathematical Physics - Tập 14 Số 12 - Trang 1817-1828 - 1973
New reductions of the Kadomtsev–Petviashvili and two-dimensional Toda lattice hierarchies via symmetry constraintsNew 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 hierarchy are also presented.
Journal of Mathematical Physics - Tập 33 Số 11 - Trang 3676-3686 - 1992
Multicomponent integrable reductions in the Kadomtsev–Petviashvilli hierarchyNew types of reductions of the Kadomtsev–Petviashvili (KP) hierarchy are considered on the basis of Sato’s approach. Within this approach the KP hierarchy is represented by infinite sets of equations for potentials u2,u3,..., of pseudodifferential operators and their eigenfunctions Ψ and adjoint eigenfunctions Ψ*. The KP hierarchy was studied under constraints of the following type (∑ni=1 ΨiΨ*i)x = Sκ,x where Sκ,x are symmetries for the KP equation and Ψi(λi), Ψ*i(λi) are eigenfunctions with eigenvalue λi. It is shown that for the first three cases κ=2,3,4 these constraints give rise to hierarchies of 1+1-dimensional commuting flows for the variables u2, Ψ1,...,Ψn, Ψ*1,...,Ψ*n. Bi-Hamiltonian structures for the new hierarchies are presented.
Journal of Mathematical Physics - Tập 34 Số 4 - Trang 1429-1446 - 1993
A homogeneous Hilbert problem for the Kinnersley–Chitre transformations of electrovac space-timesThe homogeneous Hilbert problem which we recently formulated for Kinnersley–Chitre transformations of vacuum spacetimes is here generalized to handle transformations of electrovac spacetimes. This provides in particular a simple derivation of our previously published integral equation.
Journal of Mathematical Physics - Tập 21 Số 6 - Trang 1418-1422 - 1980
A homogeneous Hilbert problem for the Kinnersley–Chitre transformationsA homogeneous Hilbert (Riemann) problem (HHP) is introduced for carrying out the Kinnersley–Chitre transformations of the set V of all axially symmetric stationary vacuum spacetimes, and the spacetimes which are like the axially symmetric stationary ones except that both Killing vectors are spacelike. A proof, which is independent of the Kinnersley–Chitre formalism, establishes that the HHP transforms the potential (for certain closed self-dual 2 forms) F0(x, t) of any given member of V into the potential F (x, t) of another member of V. Two illustrative examples involving the Minkowski space F0(x, t) are given. The representation used for the Geroch group K, the singularities and gauge of the potentials, and possible applications of the HHP are discussed.
Journal of Mathematical Physics - Tập 21 Số 5 - Trang 1126-1140 - 1980
Diffuse Transmission of Light from a Central Source through an Inhomogeneous Spherical Shell with Isotropic ScatteringA partial-differential-integral equation is derived in this paper for the angular distribution of the radiation which is diffusely transmitted through an inhomogeneous, isotropically scattering, spherical shell when there is a constant net flux of radiation normally incident on the inner surface. An equation is also derived for the strength of the diffusely reflected radiation when the shell is illuminated at each point on the outer surface by constant isotropic incident radiation.
The equations obtained appear to lend themselves well to numerical solution. Astrophysically, the situation corresponds to determining the brightness of a spherical planetary nebula. As far as is known, the equations are new and exact.
Journal of Mathematical Physics - Tập 9 Số 6 - Trang 909-912 - 1968
Exact semiclassical expansions for one-dimensional quantum oscillatorsA set of rules is given for dealing with WKB expansions in the one-dimensional analytic case, whereby such expansions are not considered as approximations but as exact encodings of wave functions, thus allowing for analytic continuation with respect to whichever parameters the potential function depends on, with an exact control of small exponential effects. These rules, which include also the case when there are double turning points, are illustrated on various examples, and applied to the study of bound state or resonance spectra. In the case of simple oscillators, it is thus shown that the Rayleigh–Schrödinger series is Borel resummable, yielding the exact energy levels. In the case of the symmetrical anharmonic oscillator, one gets a simple and rigorous justification of the Zinn-Justin quantization condition, and of its solution in terms of “multi-instanton expansions.”
Journal of Mathematical Physics - Tập 38 Số 12 - Trang 6126-6184 - 1997
Open perturbation and the Riccati equation: Algebraic determination of the quartic anharmonic oscillator energies and eigenfunctionsAn algebraic procedure is proposed for the analytical solution of Schrödinger equations that can be viewed as a factorizable equation with an additional potential V(x). Once V(x) has been expanded in a series of suitable x-basis functions u=u(x), which are specific to each factorization type, the solution of the Riccati equation associated with the given equation is performed by means of an open perturbation technique, i.e., at each order of the perturbation, an additional balance u-dependent term is introduced so that the resulting equation becomes solvable. Since the unperturbed potential involves the whole given potential and since the balance term is expected to be small, improved results are expected at low orders of the perturbation, even at the zeroth order. The procedure, well adapted to the use of computer algebra, is applied to the solution of the gx4-anharmonic oscillator equation: by means of very simple algebraic manipulations, the trend of the exact values of the energies is rather well reproduced for a large range of values of the coupling constant (g=0.002 to g=20000).
Journal of Mathematical Physics - Tập 38 Số 11 - Trang 5483-5492 - 1997
Tổng số: 181
- 1
- 2
- 3
- 4
- 5
- 6
- 10