6 M ay 1 99 9 Difficult Problems Having Easy Solutions

ar X iv : m at h / 99 05 02 6 v 1 [ m at h . A G ] 5 M ay 1 99 9 DIFFERENTIAL EQUATIONS CHARACTERISING QUASI - PERIODIC SOLUTIONS OF THE KP HIERARCHY

We compute an infinite system of differential equations that characterises the quasi-periodic solutions of the KP hierarchy ; that is, the solutions of this new hierarchy are precisely those solutions of the KP hierarchy coming from theta functions of Jacobians.

ar X iv : p at t - so l / 9 90 50 05 v 1 2 1 M ay 1 99 9 Discrete breathers in systems with homogeneous potentials - analytic solutions

We construct lattice Hamiltonians with homogeneous interaction potentials which allow for explicit breather solutions. Especially we obtain exponentially localized solutions for d-dimensional lattices with d = 2, 3.

ar X iv : a st ro - p h / 99 05 37 6 v 1 2 8 M ay 1 99 9 A 6 . 4 - hr positive superhump period in TV

v 1 2 7 M ay 1 99 9 Some exact non - vacuum Bianchi VI 0 and VII 0 instantons

We report some new exact instantons in general relativity. These solutions are Kähler and fall into the symmetry classes of Bianchi types VI0 and VII0, with matter content of a stiff fluid. The qualitative behaviour of the solutions is presented, and we compare it to the known results of the corresponding self-dual Bianchi solutions. We also give axisymmetric Bianchi VII0 solutions with an elec...

ar X iv : s ol v - in t / 9 70 50 02 v 1 1 M ay 1 99 7 Rational solutions to d - P IV

We study the rational solutions of the discrete version of Painlevé’s fourth equation (d-PIV). The solutions are generated by applying Schlesinger transformations on the seed solutions −2z and −1/z. After studying the structure of these solutions we are able to write them in a determinantal form that includes an interesting parameter shift that vanishes in the continuous limit.