Long Borel hierarchies

We show that there is a model of ZF in which the Borel hierarchy on the reals has length ω₂. This implies that ω₁ has countable cofinality, so the axiom of choice fails very badly in our model. A similar argument produces models of ZF in which the Borel hierarchy has exactly λ + 1 levels for any given limit ordinal λ less than ω₂. We also show that assuming a large cardinal hypothesis there are models of ZF in which the Borel hierarchy is arbitrarily long. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Six-dimensional heavenly equation. Dressing scheme and the hierarchy

We consider six-dimensional heavenly equation as a reduction in the framework of general six-dimensional linearly degenerate dispersionless hierarchy. We characterise the reduction in terms of wave functions, introduce generating relation, Lax–Sato equations and develop the dressing scheme for the reduced hierarchy. Using the dressing scheme, we construct a class of solutions for six-dimensional heavenly equation in terms of implicit functions.     

Soliton Solutions of Integrable Hierarchies and Coulomb Plasmas

Some direct relations are given between soliton solutions of integrable hierarchies and thermodynamic quantities of the Coulomb plasmas on the plane. We find that certain soliton solutions of the Kadomtsev–Petviashvili (KP) and B-type KP (BKP) hierarchies describe 2D one- or two-component lattice plasmas at special boundary conditions and fixed temperatures. It is shown that different reductions of integrable hierarchies describe pure or dipole Coulomb gases on 1D submanifolds embedded in the 2D space.     

Additional symmetries and string equations of the noncommutative B and C type KP hierarchies

In this paper, we construct the noncommutative B and C type KP hierarchies using pseudo-differential operators and reducing conditions. Further a series of additional flows of the noncommutative B and C type KP hierarchies will be defined and the additional symmetries constitute the B and C type infinite dimensional Lie algebra W_1+∞. In addition, the generating function of the additional symmetries can also be proved to have a nice form in terms of wave functions. Further, the string equations of the noncommutative B and C type KP hierarchies are derived.     

A Uniform Approach to Fundamental Sequences and Hierarchies

In this article we give a unifying approach to the theory of fundamental sequences and their related Hardy hierarchies of number-theoretic functions and we show the equivalence of the new approach with the classical one. Mathematics Subject Classification: 03D20, 03F15, 03E10.     

Huygens' Principle,Dirac Operators,and Rational Solutions of the AKNS Hierarchy

We prove that rational solutions of the AKNS hierarchy of the form q=σ/τ and r=ρ/τ, where (σ,τ,ρ) are certain Schur functions, naturally yield Dirac operators of strict Huygens' type, i.e., the support of their fundamental solutions is the surface of the light-cone. This strengthens the connection between the theory of completely integrable systems and Huygens' principle by extending to the Dirac operators and the rational solutions of the AKNS hierarchy a classical result of Lagnese and Stellmacher concerning perturbations of wave operators. Mathematics Subject Classifications (2000) 37K10, 35Qxx, 35B40.     

Integrable Systems and Rank-One Conditions for Rectangular Matrices

We give a determinantal formula for tau functions of the KP hierarchy in terms of rectangular constant matrices A, B, and C satisfying a rank-one condition. This result is shown to generalize and unify many previous results of different authors on constructions of tau functions for differential and difference integrable systems from square matrices satisfying rank-one conditions. In particular, its explicit special cases include Wilson's formula for tau functions of the rational KP solutions in terms of Calogero–Moser Lax matrices and our previous formula for the KP tau functions in terms of almost-intertwining matrices.     

A PDE for non-intersecting Brownian motions and applications

Consider N=n₁+n₂+?+np non-intersecting Brownian motions on the real line, starting from the origin at t=0, with ni particles forced to reach p distinct target points βi at time t=1, with β₁<β₂<?<βp. This can be viewed as a diffusion process in a sector of RN. This work shows that the transition probability, that is the probability for the particles to pass through windows at times tk, satisfies, in a new set of variables, a non-linear PDE which can be expressed as a near-Wronskian; that is a determinant of a matrix of size p+1, with each row being a derivative of the previous, except for the last column. It is an interesting open question to understand those equations from a more probabilistic point of view.As an application of these equations, let the number of particles forced to the extreme points β₁ and βp tend to infinity; keep the number of particles forced to intermediate points fixed (inliers), but let the target points themselves go to infinity according to a proper scale. A new critical process appears at the point of bifurcation, where the bulk of the particles forced to depart from those going to . These statistical fluctuations near that point of bifurcation are specified by a kernel, which is a rational perturbation of the Pearcey kernel. This work also shows that such equations are an essential tool in obtaining certain asymptotic results. Finally, the paper contains a conjecture.     

Bäcklund transformations for new fourth Painlevé hierarchies

We consider a system of equations defined using the Hamiltonian operator of the Boussinesq hierarchy, as well as two successive modifications thereof. We are able to reduce the order of these three systems and give Bäcklund transformations between the integrated equations. We also give auto-Bäcklund transformations for the two modified systems.Particular cases of two of the three equations considered correspond to generalized fourth Painlevé hierarchies and are new; these are particular cases of the two modified systems. Thus we obtain auto-Bäcklund transformations for these new fourth Painlevé hierarchies, as well as Bäcklund transformations between our hierarchies. Our results on reduction of order are also applicable in this special case, and include as a particular example a reduction of order for the scaling similarity reduction of the Boussinesq equation, a result which, remarkably, seems not to have been given previously.