Monopoles and Baker functions

The work in this paper pertains to the solutions of Nahm's equations, which arise in the Atiyah-Drinfield-Hitchin-Manin-Nahm construction of solutions to the Bogomol'nyi equations for static monopoles. This paper provides an explicit construction of the solution of Nahm's equations which satisfy regularity and reality conditions. The Lax form of Nahm's equations is reduced to a standard eigenvalue problem by a special gauge transformation. These equations may then be solved by the method of Baker-Krichever. This leads to a compact representation of the solutions of Nahm's equations. The regularity condition is shown to be related to the monodromy of the gauge reduced linear operator. Hitchin showed that the solutions of Nahm's equations can be characterized by an algebraic curve and some data on that curve. Here, this characterization reduces to a transcendental equation involving certain loop integrals of a meromorphic differential. Donaldson coordinatized the moduli space ofk-monopoles by a class of rational maps from the Riemann sphere to itself. The data of a Baker function is equivalent to this map. This method gives an apriori construction of the (known) two monopole solutions. We also give a generalization of the two monopole solution to a class of elliptic solutions of arbitrary charge. These solutions correspond to reducible curves with elliptic components and the associated Donaldson rational function has a simple partial fraction expansion.Supported in part by the National Science Foundation, Grant Number DMS-8701318 and the Arizona Center for Mathematical Sciences, sponsored by AFOSR Contract F49620-86-C0130 with the University Research Initiative Program at the University of Arizona     

Pseudodifferential Operators of Several Variables and Baker Functions

The KP hierarchy consists of an infinite system of nonlinear partial differential equations and is determined by Lax equations, which can be constructed using pseudodifferential operators. The KP hierarchy and the associated Lax equations can be generalized by using pseudodifferential operators of several variables. We construct Baker functions associated to those generalized Lax equations of several variables and prove some of the properties satisfied by such functions.     

Entropy evolution for the Baker map

Gibbs entropy is invariant for the Baker map. A Jordan basis spectral decomposition of the Baker Frobenius-Perron operator suggests that any initial density evolves to the stationary density that has maximal entropy. This entropy conundrum is resolved by considering the difference between weak and strong convergence. A binary representation is used to make these points transparent. (c) 1998 American Institute of Physics.     

Transient Chaos and Critical States in Generalized Baker Maps

Generalized multibaker maps are introduced to model dissipative systems which are spatially extended only in certain directions and escape of particles is allowed in other ones. Effects of nonlinearity are investigated by varying a control parameter. Emphasis is put on the appearance of the critical state representing the borderline of transient chaos, where anomalous behavior sets in. The investigations extend to the conditionally invariant and the related natural measures and to transient diffusion in normal and critical states as well. Permanent chaos is also considered as a special case.