Darboux evaluations of algebraic Gauss hypergeometric functions

This paper presents explicit expressions for algebraic Gauss hypergeometric functions. We consider solutions of hypergeometric equations with the tetrahedral, octahedral and icosahedral monodromy groups. Conceptually, we pull-back such a hypergeometric equation onto its Darboux curve so that the pull-backed equation has a cyclic monodromy group. Minimal degree of the pull-back coverings is 4, 6 or 12 (for the three monodromy groups, respectively). In explicit terms, we replace the independent variable by a rational function of degree 4, 6 or 12, and transform hypergeometric functions to radical functions.Comment: The list of seed hypergeometric evaluations (in Section 2) reduced by half; uniqueness claims explained; 34 pages; Kyushu Journal of Mathematics, 201

Computing the differential Galois group of a parameterized second-order linear differential equation

We develop algorithms to compute the differential Galois group $G$ associated to a parameterized second-order homogeneous linear differential equation of the form $$\tfrac{\partial^2}{\partial x^2} Y + r_1 \tfrac{\partial}{\partial x} Y + r_0 Y = 0,$$ where the coefficients $r_1, r_0 \in F(x)$ are rational functions in $x$ with coefficients in a partial differential field $F$ of characteristic zero. Our work relies on the procedure developed by Dreyfus to compute $G$ under the assumption that $r_1 = 0$. We show how to complete this procedure to cover the cases where $r_1 \neq 0$, by reinterpreting a classical change of variables procedure in Galois-theoretic terms.Comment: 14 page

The Painlev\'e methods

This short review is an introduction to a great variety of methods, the collection of which is called the Painlev\'e analysis, intended at producing all kinds of exact (as opposed to perturbative) results on nonlinear equations, whether ordinary, partial, or discrete.Comment: LaTex 2e, subject index, Nonlinear integrable systems: classical and quantum, ed. A. Kundu, Special issue, Proceedings of Indian Science Academy,

Differential transcendence criteria for second-order linear difference equations and elliptic hypergeometric functions

We develop general criteria that ensure that any non-zero solution of a given second-order difference equation is differentially transcendental, which apply uniformly in particular cases of interest, such as shift difference equations, q-dilation difference equations, Mahler difference equations, and elliptic difference equations. These criteria are obtained as an application of differential Galois theory for difference equations. We apply our criteria to prove a new result to the effect that most elliptic hypergeometric functions are differentially transcendental

Painleve versus Fuchs

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}.Comment: 18 pages, Dedicated to the centenary of the publication of the Painleve VI equation in the Comptes Rendus de l'Academie des Sciences de Paris by Richard Fuchs in 190

Solitary waves of nonlinear nonintegrable equations

Our goal is to find closed form analytic expressions for the solitary waves of nonlinear nonintegrable partial differential equations. The suitable methods, which can only be nonperturbative, are classified in two classes. In the first class, which includes the well known so-called truncation methods, one \textit{a priori} assumes a given class of expressions (polynomials, etc) for the unknown solution; the involved work can easily be done by hand but all solutions outside the given class are surely missed. In the second class, instead of searching an expression for the solution, one builds an intermediate, equivalent information, namely the \textit{first order} autonomous ODE satisfied by the solitary wave; in principle, no solution can be missed, but the involved work requires computer algebra. We present the application to the cubic and quintic complex one-dimensional Ginzburg-Landau equations, and to the Kuramoto-Sivashinsky equation.Comment: 28 pages, chapter in book "Dissipative solitons", ed. Akhmediev, to appea

A Note on Fractional KdV Hierarchies

We introduce a hierarchy of mutually commuting dynamical systems on a finite number of Laurent series. This hierarchy can be seen as a prolongation of the KP hierarchy, or a ``reduction'' in which the space coordinate is identified with an arbitrarily chosen time of a bigger dynamical system. Fractional KdV hierarchies are gotten by means of further reductions, obtained by constraining the Laurent series. The case of sl(3)^2 and its bihamiltonian structure are discussed in detail.Comment: Final version to appear in J. Math. Phys. Some changes in the order of presentation, with more emphasis on the geometrical picture. One figure added (using epsf.sty). 30 pages, Late