Physics – Mathematical Physics
Scientific paper
2006-02-03
J.Phys.A39:12245-12264,2006
Physics
Mathematical Physics
18 pages, Dedicated to the centenary of the publication of the Painleve VI equation in the Comptes Rendus de l'Academie des Sc
Scientific paper
10.1088/0305-4470/39/39/S16
The sigma form of the Painlev{\'e} VI equation contains four arbitrary parameters and generically the solutions can be said to be genuinely ``nonlinear'' because they do not satisfy linear differential equations of finite order. However, when there are certain restrictions on the four parameters there exist one parameter families of solutions which do satisfy (Fuchsian) differential equations of finite order. We here study this phenomena of Fuchsian solutions to the Painlev{\'e} equation with a focus on the particular PVI equation which is satisfied by the diagonal correlation function C(N,N) of the Ising model. We obtain Fuchsian equations of order $N+1$ for C(N,N) and show that the equation for C(N,N) is equivalent to the $N^{th}$ symmetric power of the equation for the elliptic integral $E$. We show that these Fuchsian equations correspond to rational algebraic curves with an additional Riccati structure and we show that the Malmquist Hamiltonian $p,q$ variables are rational functions in complete elliptic integrals. Fuchsian equations for off diagonal correlations $C(N,M)$ are given which extend our considerations to discrete generalizations of Painlev{\'e}.
Boukraa Salah
Hassani Samira
Maillard Jean-Marie
McCoy Barry M.
Weil Jacques-Arthur
No associations
LandOfFree
Painleve versus Fuchs does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Painleve versus Fuchs, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Painleve versus Fuchs will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-485150