On the density of the Ward ansätze in the space of anti-self-dual Yang Mills solutions

A general patching matrixP for the twistor construction of antiself-dual Yang-Mills fields is approximated by a terminating Laurent series. The approximate patching matrixP(^m) is triangularized (so that it becomes one of the Ward ansätze) and the associated Riemann-Hilbert problem is solved, thereby generating an anti-self-dual solution of the Yang-Mills equations. The approximate patching matrices and the associated (exact) anti-self-dual Yang-Mills solutions are then shown to converge onP and its corresponding solution so that the Ward ansätze forms a dense subset in the solution space in the Weierstrass sense.     

Letter: Parametrization of the Kerr-NUT Solution

The dragging of the Kerr-NUT solution does not tend to zero at infinity. To modify this solution in order to produce a good asymptotic behaviour we transform it by introducing two further parameters with the aid of a SU(1,1) transformation followed by a unitary transformation. By imposing a certain relation between these parameters we obtain a new solution with a good asymptotic behaviour for any value of l, the NUT parameter. The new solution corresponds to a parametrized Kerr solution and we show that l is linked to the form of its ergosphere.     

On local solutions of the initial value problem for the Vlasov-Maxwell equation

The initial value problem of the Vlasov-Maxwell equation has a unique solution in a time interval [0,T] for each initial data in some function space.T is estimated by the size of the initial data. The solution is classical, if the initial data is smooth.     

On the Bardeen-Cooper-Schrieffer integral equation in the theory of superconductivity

The Bardeen-Cooper-Schrieffer integral equation with a positive kernel is studied in full generality. It is shown that, there exists a unique finite transition temperature, T _c so that, if T _c ,the equation possesses a positive solution, representing the onset of the superconducting phase, while if T>T _c ,the only solution of the equation is the trivial one, indicating the occurrence of the normal phase. Moreover, it is demonstrated that such a positive solution may be approximated by a sequence of solutions of the equation restricted on bounded domains. This latter result provides a useful computational scheme for the problem.     

Unconventional spin solutions of the Schwinger and Thirring models

A satisfactory operator solution of the Schwinger-Thirring model is found which ase→0 org→0 tends respectively to the operator solution of the Thirring or the Schwinger model. We also propose the general solutions of the Schwinger-Thirring model which can be associated with the solution of the Thirring model with unconventional statistics. In particular our method allows for a generalization of the Schwinger model for arbitrary spins giving rise to a new solution with the “photon” mass dependent ons. The following consequences for a bosonization procedure is also described.

     

From the Solution of the Tsarev System to the Solution of the Whitham Equations

We study the Cauchy problem for the Whitham modulation equations for increasing smooth initial data. The Whitham equations are a collection of one-dimensional quasi-linear hyperbolic systems. This collection of systems is enumerated by the genus g=0,1,2, ... of the corresponding hyperelliptic Riemann surface. Each of these systems can be integrated by the so-called hodograph transformation introduced by Tsarev. A key step in the integration process is the solution of the Tsarev linear overdetermined system. For each g>0, we construct the unique solution of the Tsarev system, which matches the genus g+1 and g–1 solutions on the transition boundaries.     

The evolution of the cluster size distribution in a coagulation system

The coagulation equation with kernelK _ij =A+B(i+j)+C _ij and arbitrary initial conditions is studied analytically and a simple expression for the solution is found. For monodisperse initial conditions, we recover the known size distribution expressed in terms of a degeneracy factorN _k, which is determined by a recursion relation. For polydisperse initial conditions, a similar solution form is found, which includes a degeneracy factorN _kl, also determined by a recursion relation. The physical meaning ofN _kl and the recursion relation is given. A method to get explicit expressions forN _k andN _kl is illustrated. Finally, the pre-gel solution is given explicitly and a general method to get the post-gel solution is proposed.     

On the Uniqueness of the Solution to the Three-Body Problem with Zero-Range Interactions

We consider the uniqueness of the solution to a three-body problem with zero-range Skyrme interactions in configuration space. With the lowest, k⁰, two-body term alone the problem is known to have no unique solution as the system collapses – the variational estimate of the energy tends towards negative infinity, the size of the system towards zero. We argue that the next, k², two-body term removes the collapse and the three-body system acquires finite ground-state energy and size. The three-body interaction term is thus not necessary to provide a unique solution to the problem.     

The series solution of the Marchenko equation

The convergence of a Neumann-series solution of the Marchenko equation in L ₂[x, ∞) is proved and an explicit expression of the series solution is given.     

Some features of the temperature dependence of the hypersonic velocity in an aqueous solution of tertiary butyl alcohol

The dependence of the hypersonic velocity V in an aqueous solution of tertiary butyl alcohol on the solution temperature t has been measured from the Mandelshtam-Brillouin scattering spectra. A change in the temperature derivative of the velocity, dV/dt, in a narrow range from ≈43 to ≈44°C, with jumps at its boundaries, is revealed. The measurement results and their comparison with the previous data indicate the existence of structural transitions with respect to temperature and concentration in these solutions.     

Solution of the Fokker-Planck equation for the shock wave problem

An eigenexpansion solution of the time-independent Brownian motion Fokker-Planck equation is given for a situation in which the external acceleration is a step function. The solution describes the heavy-species velocity distribution function in a binary mixture undergoing a shock wave, in the limit of high dilution of the heavy species and negligible width of the light-gas internal shock. The diffusion solution is part of the eigenexpansion. The coefficients of the series of eigenfunctions are obtained analytically with transcendentally small errors of order exp(–1/M), whereM 1 is the mass ratio. Comparison is made with results from a hypersonic approximation.     

On the general form of the Beltrami equation and Papkovich's solution of the axially symmetrical problem of the classical theory of elasticity

The paper shows the connection between the general form of the Beltrami equation of compatibility, which had already been derived by the author, and the form given for it in orthogonal curvilinear coordinates by Lurye (para 2, 3).Some of the properties of Papkovich's general solution of the axially symmetrical problem (without body forces) of the theory of elasticity and its relation to the Finzi-Krutkov solution are discussed so as to supplement to a certain extent the paper of Trenin (para 4, 5, 6).     

An explicit one-soliton solution of the (2+1)-dimensional generalized sine-gordon equations

The (n+1)-dimensional differential geometric generalization of the sine-Gordon equation (SGE) given by Tenenblat and Terng is solved explicitly in the casen=2 to obtain a one-soliton solution. The solution yields the soliton solution of the (1+1)-dimensional SGE in the limit as one of the three independent variables approaches infinity. However, more than one variable plays the role of time in these limits.     

Solution of the Bethe-Goldstone equation for the reaction matrix in finite nuclei

A method for solving the BG equation for the reaction matrixt in finite nuclei is presented. The application of this method is demonstrated for a one-dimensional case, which is similar to the problem where the internucleon potential acts only in the relatives-state. The single particle potential has a harmonic oscillator form and the phenomenological internucleon potentialv(r) contains a hard core and an attractive part of the Yukawa type. By taking the exclusion principle into account exactly an infinite system of integral equations is obtained. It is proved that the solution of the corresponding finite system converges to the exact solution. An iteration method for solving such a finite system with an arbitrary number of equations is developed. Its main feature consists in the exclusion of the dependence on the hard core part ofv(r) (which is treated as the limit case of a rectangular repulsive potential with a variable heightv ₀). This exclusion transforms the original system to a system of integral equations depending only on the attractive part ofv(r) and to a linear algebraic system. Both these systems can be solved by iteration for all values ofv ₀ as well as for v₀= +. The numerical results confirm the rapid convergence of the proposed iteration method and demonstrate that the solution of the finite system with a sufficiently large number of equations approximates the exact solution very precisely.     

Factorized S-matrices in two dimensions as the exact solutions of certain relativistic quantum field theory models

The general properties of the factorized S-matrix in two-dimensional space-time are considered. The relation between the factorization property of the scattering theory and the infinite number of conservation laws of the underlying field theory is discussed. The factorization of the total S-matrix is shown to impose hard restrictions on two-particle matrix elements: they should satisfy special identities, the so-called factorization equations. The general solution of the unitarity, crossing and factorization equations is found for the S-matrices having isotopic O(N)-symmetry. The solution turns out to have different properties for the cases N = 2 and N 3. For N = 2 the general solution depends on one parameter (of coupling constant type), whereas the solution for N 3 has no parameters but depends analytically on N. The solution for N = 2 is shown to be an exact soliton S-matrix of the sine-Gordon model (equivalently the massive Thirring model). The total S-matrix of the model is constructed. In the case of N 3 there are two “minimum” solutions, i.e., those having a minimum set of singularities. One of them is shown to be an exact S matrix of the quantum O(N)-symmetric nonlinear σ-model, the other is argued to describe the scattering of elementary particles of the Gross-Neveu model.     

Classical solution of the nonlinear Boltzmann equation in allR 3: Asymptotic behavior of solutions

Proof is given of the existence of a classical solution to the nonlinear Boltzmann equation in allR ³. The solution, which is global in time, exists if the initial data go to zero fast enough at infinity and the mean free path is sufficiently large. The solution is smooth in the space variable if the initial value is smooth. The asymptotic behavior of solutions is also given. It is shown that ast the solution to the Boltzmann equation can be approximated by the solution to the free motion problem.     

Asymptotic Stability of the Stationary Solution to the Compressible Navier–Stokes Equations in the Half Space

We investigate the existence and the asymptotic stability of a stationary solution to the initial boundary value problem for the compressible Navier–Stokes equation in a half space. The main concern is to analyze the phenomena that happens when the fluid blows out through the boundary. Thus, it is natural to consider the problem in the Eulerian coordinate. We have obtained the two results for this problem. The first result is concerning the existence of the stationary solution. We present the necessary and sufficient condition which ensures the existence of the stationary solution. Then it is shown that the stationary solution is time asymptotically stable if an initial perturbation is small in the suitable Sobolev space. The second result is proved by using an L²-energy method with the aid of the Poincaré type inequality.The second author's work was supported in part by Grant-in-Aid for Scientific Research (C)(2) 14540200 of the Ministry of Education, Culture, Sports, Science and Technology and the third author's work was supported by JSPS postdoctoral fellowship under P99217.     

Resonance structure of the beta-strength function

The beta-decay strength function S _β(E) is described within two approaches: numerical solution of equation of the theory of finite Fermi systems for the effective nuclear field and solution of these equations using the quasi-classical approximation. Calculations were carried out for the isotopes ⁷¹Ge and ¹²⁷Xe. A comparison with experimental data showed their good accuracy. The resonance structure of the function S _β(E) and the quenching-effect resulting from the effective charge of quasiparticles in the nucleus are discussed.     

ExactS-matrix of the adjoint SU(N) representation

We have calculated the exact factorisedS-matrices of the adjoint SU(N) representation in 1+1 space-time dimensions. Besides the trivial solution the only realised solution exhibits anO(N ²–1) symmetry.     

Oscillations and concentrations in weak solutions of the incompressible fluid equations

The authors introduce a new concept of measure-valued solution for the 3-D incompressible Euler equations in order to incorporate the complex phenomena present in limits of approximate solutions of these equations. One application of the concepts developed here is the following important result: a sequence of Leray-Hopf weak solutions of the Navier-Stokes equations converges in the high Reynolds number limit to a measure-valued solution of 3-D Euler defined for all positive times. The authors present several explicit examples of solution sequences for 3-D incompressible Euler with uniformly bounded local kinetic energy which exhibit complex phenomena involving both persistence of oscillations and development of concentrations. An extensions of the concept of Young measure is developed to incorporate these complex phenomena in the measure-valued solutions constructed here.Partially supported by N.S.F. GrantPartially supported by N.S.F. Grant 84-0223 and 86-11110