Orthogonal polynomial random matrix models of N x N hermitian matrices lead to Fredholm determinants of integral operators with kernel of the form (φ(x)φ(y) — ψ(x)φ(y))/x — y. This paper is concerned with the Fredholm determinants of integral operators having kernel of this form and where the underlying set is the union of intervals J= [JTM=1 (βij-u Λ2 J ) The emphasis is on the determinants th...

is a physically important nonlinear Galilean-invariant system describing slow changes in linear waves due to a combination of nonlinearity, dispersion and dissipative effects [1]. When the dissipation is dominant, (1) is known to possess shock-like solitary waves [2] while in the pure dispersive limit (y = 0), (1) admits the well known K-dV solitons [3]. In the case of weak dissipation in (1), ...

We give a Hamiltonian analysis of the asymptotically flat spherically symmetric system of gravity coupled to a scalar field. This 1+1 dimensional field theory may be viewed as the ”standard model” for studying black hole physics. Our analysis is adapted to the flat slice Painleve-Gullstrand coordinates. We give a Hamiltonian action principle for this system, which yields an asymptotic mass form...

We present some observations on the asymptotic behaviour of coefficients in Laurent series expansion solutions first Painleve equation. For general solution, explicit recursive formulae for Taylor tau-function around a zero are given, which natural extensions analogous elliptic sigma function, as given by Weierstrass. Numerical and exact results symmetric solution is singular at origin also pre...