Algebro-Geometric Solution of the Discrete KP Equation over a Finite Field out of a Hyperelliptic Curve

We transfer the algebro-geometric method of construction of solutions of the discrete KP equation to the finite field case. We emphasize the role of the Jacobian of the underlying algebraic curve in construction of the solutions. We illustrate in detail the procedure on example of a hyperelliptic curve.Acknowledgement The paper was partially supported by the University of Warmia and Mazury in Olsztyn under the grant 522-1307-0201 and by KBN grant 2 P03B 12622.     

On multivortices in the electroweak theory II: Existence of Bogomol'nyi solutions in ℝ2

In this paper we study the Bogomol'nyi equations of the electroweak theory in the full plane. We will show that, for any distribution of the vortices, there exists a two parameter family of gauge-distinct solutions. Moreover, we also establish some sharp decay rate estimates for these solutions.Research supported in part by NSF grant DMS-88-02858 and DOE grant DE-FG02-86ER250125     

Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping

We consider a model of hyperbolic conservation laws with damping and show that the solutions tend to those of a nonlinear parabolic equation time-asymptotically. The hyperbolic model may be viewed as isentropic Euler equations with friction term added to the momentum equation to model gas flow through a porous media. In this case our result justifies Darcy's law time-asymptotically. Our model may also be viewed as an elastic model with damping.Research supported in part by Energy Dept. grant DEFG 02-88-ER25053Research supported in part by NSF grant DMS 90-0226 and Army grant DAAL 03-91-G0017     

Unusual direction of waves in the piston model for tsunami

Asymptotic solutions of the tsunami piston model with initial data in the form of Gaussian exponentials with oscillations are considered. The direction of waves described by these solutions is studied. Initial conditions are found that excite waves of unusual direction. The result is confirmed numerically. This work was supported by the Russian Foundation for Basic Research under grant no.05-01-00968.     

Action minima among solutions to a class of Euclidean scalar field equations

We show that for a wide class of Euclidean scalar field equations, there exist non-trivial solutions, and the non-trivial solution of lowest action is spherically symmetric. This fills a gap in a recent analysis of vacuum decay by one of us.Work supported in part by the US National Science Foundation under grant Nr. PHY75-20427     

The large time stability of sound waves

We demonstrate the existence of solutions to the full 3×3 system of compressible Euler equations in one space dimension, up to an arbitrary timeT>0, in the case when the initial data has arbitrarily large total variation, and sufficiently small supnorm. The result applies to periodic solutions of the Euler equations, a nonlinear model for sound wave propagation in gas dynamics. Our analysis establishes a growth rate for the total variation that depends on a new length scaled that we identify in the problem. This length scale plays no role in 2×2 systems, (or any system possessing a full set of Riemann coordinates), nor in the small total variation problem forn×n systems, the cases originally addressed by Glimm in 1965. Recent work by a number of authors has demonstrated that when the total variation is sufficiently large, solutions of 3×3 systems of conservation laws can in general blow up in finite time, (independent of the supnorm), due to amplifying instabilities created by the non-trivial Lie algebra of the vector fields that define the elementary waves. For the large total variation problem, there is an interaction between large scale effects that amplify and small scale effects that are stable, and we show that the length scale on which this interaction occurs isd. In the limitd, we recover Glimm's theorem, and we observe that there exist linearly degenerate systems within the class considered for which the growth rate we obtain is sharp.Supported in part by NSF Applied Mathematics grant numbers DMS-92-06631, DMS-95000694, in part by ONR, US Navy grant number N00014-94-1-0691, A Guggenheim fellowship, and by the Institute of Theoretical Dynamics, UC-Davis.Partially supported by DOE grant number DE-FG02-88ER25053 while at the Courant Institute, and by NSF grant number DMS-9201581 and DOE grant number DE-FG02-90ER25084.     

An Integral Spectral Representation of the Propagator for the Wave Equation in the Kerr Geometry

We consider the scalar wave equation in the Kerr geometry for Cauchy data which is smooth and compactly supported outside the event horizon. We derive an integral representation which expresses the solution as a superposition of solutions of the radial and angular ODEs which arise in the separation of variables. In particular, we prove completeness of the solutions of the separated ODEs. This integral representation is a suitable starting point for a detailed analysis of the long-time dynamics of scalar waves in the Kerr geometry. Research supported by NSERC grant # RGPIN 105490-2004. Research supported in part by the NSF, Grant No. DMS-010-3998. Research supported in part by the NSF, Grant No. 33-585-7510-2-30.     

Scattering for the wave equation on the Schwarzschild Metric

We study the wave equation for the Schwarzschild metric. Wave operators are constructed which yield solutions with given asymptotic behavior either at infinity or on the horizon. We prove asymptotic completeness for these wave operators.Supported by NSF grant No. PHY82-204399.     

Global finite-energy solutions of the Maxwell-Schrödinger system

The existence of global finite-energy solutions is proved for the initial value problem for the Maxwell-Schrödinger system in the Coulomb, Lorentz and temporal gaugesThis research was done primarily while all three authors were at Brown University. It was supported in part by NSF grant 90-23864 and ARO grant DAAH 04-93-G-0198.     

Stability of shock waves for the Broadwell equations

For the Broadwell model of the nonlinear Boltzmann equation, there are shock profile solutions, i.e. smooth traveling waves that connect two equilibrium states. For weak shock waves, we prove asymptotic (in time) stability with respect to small perturbations of the initial data. Following the work of Liu [7] on shock wave stability for viscous conservation laws, the method consists of analyzing the solution as the sum of a shock wave, a diffusive wave, a linear hyperbolic wave and an error term. The diffusive and linear hyperbolic waves are approximate solutions of the fluid dynamic equations corresponding to the Broadwell model. The error term is estimated using a variation of the energy estimates of Kawashima and Matsumura [6] and the characteristic energy method of Liu [7].Research supported by the Office of Naval Research through grant N00014-81-0002 and by the National Science Foundation through grant NSF-MCS-83-01260Research supported by the National Science Foundation through grant DMS-84-01355     

Discrete Miura Opers and Solutions of the Bethe Ansatz Equations

Solutions of the Bethe ansatz equations associated to the XXX model of a simple Lie algebra come in families called the populations. We prove that a population is isomorphic to the flag variety of the Langlands dual Lie algebra The proof is based on the correspondence between the solutions of the Bethe ansatz equations and special difference operators which we call the discrete Miura opers. The notion of a discrete Miura oper is one of the main results of the paper.For a discrete Miura oper D, associated to a point of a population, we show that all solutions of the difference equation DY=0 are rational functions, and the solutions can be written explicitly in terms of points composing the population.Supported in part by NSF grant DMS-0140460Supported in part by NSF grant DMS-0244579     

Stationary Dyonic regular and black hole solutions

We consider globally regular and black hole solutions in SU(2) Einstein–Yang–Mills–Higgs theory, coupled to a dilaton field. The basic solutions represent magnetic monopoles, monopole–antimonopole systems or black holes with monopole or dipole hair. When the globally regular solutions carry additionally electric charge, an angular momentum density results, except in the simplest spherically symmetric case. We evaluate the global charges of the solutions and their effective action, and analyze their dependence on the gravitational coupling strength. We show, that in the presence of a dilaton field, the black hole solutions satisfy a generalized Smarr type mass formula. B. Kleihaus gratefully acknowledges support by the German Aerospace Center. F. Navarro-Lérida gratefully acknowledges support by the Ministerio de Educación y Ciencia under grant EX2005-0078.     

Ergosphere instability

We consider stationary asymptotically flat spacetimes having an ergosphere but with no horizon. In the framework of linear perturbation theory such configurations are unstable or marginally unstable to scalar and electromagnetic perturbations.Research supported in part by the National Science Foundation under grant MPS 74-17456 with the University of Chicago and grant MPS 74-7456 at the University of Wisconsin-Milwaukee     

Isgur-Wise functions for confined light quarks in a colour electric potential

We explore the influence on the Isgur-Wise function of the colour electric potential between heavy and light quarks in mesons. It is shown that in bag models, its inclusion tends to restore light quark flavour symmetry relative to the MIT bag predictions, and that relative to this model it flattens the Isgur-Wise function. Results compare very well with observations.Work supported in part (M.S.) by the KBN grant no. 20-38-09101. We also thank the Norwegian Research Council for a travel grant     

Decay estimates for Schrödinger equations

We prove global existence and optimal decay estimates for classical solutions with small initial data for nonlinear nonlocal Schrödinger equations. The Laplacian in the Schrödinger equation can be replaced by an operator corresponding to a non-degenerate quadratic form of arbitrary signature. In particular, the Davey-Stewartson system is included in the the class of equations we discuss.Partially supported by NSF grant DMS-860-2031. Sloan Research Fellow     

On ergodic properties for harmonic crystals

We consider the dynamics of a harmonic crystal in n dimensions with d components, where d and n are arbitrary, d, n ⩾ 1. The initial data are given by a random function with finite mean energy density which also satisfies a Rosenblatt-or Ibragimov-type mixing condition. The random function is close to diverse space-homogeneous processes as x _n → ±∞, with the distributions μ±. We prove that the phase flow is mixing with respect to the limit measure of statistical solutions. Partially supported by RFBR under grant no. 06-01-00096.     

Pure Point Spectrum of the Floquet Hamiltonian for the Quantum Harmonic Oscillator Under Time Quasi-Periodic Perturbations

We prove that the 1-d quantum harmonic oscillator is stable under spatially localized, time quasi-periodic perturbations on a set of Diophantine frequencies of positive measure. This proves a conjecture raised by Enss-Veselic in their 1983 paper [EV] in the general quasi-periodic setting. The motivation of the present paper also comes from construction of quasi-periodic solutions for the corresponding nonlinear equation. Partially supported by NSF grant DMS-05-03563.     

Quantization ofSL(2,R) Chern-Simons theory

In this paper we consider the bosonic sector of the electroweak theory. It has been shown in the work of Ambjorn and Olesen that when the Higgs mass equals to the mass of theZ boson, the model in two dimensions subject to the 't Hooft periodic boundary condition may be reduced to a Bogomol'nyi system and that the solutions of the system are vortices in a dual superconductor. We shall prove using a constrained variational reformulation of the problem the existence of such vortices. Our conditions for the existence of solutions are necessary and sufficient when the vortex numberN=1,2.Research supported in part by NSF grant DMS-88-02858 and DOE grant DE-FG02-86ER250125     

Universality in the Thomas-Fermi-von Weizsäcker model of atoms and molecules

We study the Thomas-Fermi-von Weizsäcker theory of atoms and molecules. The main result is to prove universality of the structure of very large atoms and molecules, i.e., proving that the structure converges as the nuclear charges go to infinity. Furthermore we uniquely characterize the limit density as the solution to a renormalized TFW-equation. This is achieved by characterizing the strong singularities of solutions to the non-linear TFW-system.Supported by a Danish Research Academy Fellowship and U.S. National Science Foundation grant PHY-85-15288-A03     

Decay of Solutions of the Wave Equation in the Kerr Geometry

We consider the Cauchy problem for the massless scalar wave equation in the Kerr geometry for smooth initial data compactly supported outside the event horizon. We prove that the solutions decay in time in L ^∞ _loc. The proof is based on a representation of the solution as an infinite sum over the angular momentum modes, each of which is an integral of the energy variable ω on the real line. This integral representation involves solutions of the radial and angular ODEs which arise in the separation of variables. Research supported in part by the Deutsche Forschungsgemeinschaft. Research supported by NSERC grant #RGPIN 105490-2004. Research supported in part by the NSF, Grant No. DMS-010-3998. Research supported in part by the NSF, Grant No. 33-585-7510-2-30. An erratum to this article is available at .