Nonlocal representation of the sl(2, R) algebra for the Chazy equation

A demonstration of how the point symmetries of the Chazy equation become nonlocal symmetries for the reduced equation is discussed. Moreover we construct an equivalent third-order differential equation which is related to the Chazy equation under a generalized transformation, and find the point symmetries of the Chazy equation are generalized symmetries for the new equation. With the use of singularity analysis and a simple coordinate transformation we construct a solution for the Chazy equation which is given by a right Painlevé series. The singularity analysis is applied to the new third-order equation and we find that it admits two solutions, one given by a left Painlevé series and one given by a right Painlevé series where the leading-order behaviors and the resonances are explicitly those of the Chazy equation.     

Nonlinear differential equations satisfied by certain classical modular forms

A unified treatment is given of low-weight modular forms on ?? ₀(N), N = 2,3,4, that have Eisenstein series representations. For each N, certain weight-1 forms are shown to satisfy a coupled system of nonlinear differential equations, which yields a single nonlinear third-order equation, called a generalized Chazy equation. As byproducts, a table of divisor function and theta identities is generated by means of q-expansions, and a transformation law under ?? ₀(4) for the second complete elliptic integral is derived. More generally, it is shown how Picard?CFuchs equations of triangle subgroups of PSL(2, R), which are hypergeometric equations, yield systems of nonlinear equations for weight-1 forms, and generalized Chazy equations. Each triangle group commensurable with ??(1) is treated.     

The Symplectic Matrix Riccati System and a discrete form of an equation of the Chazy XII classification

An example of a non-linear third order differential equation in the Chazy XII classification is shown to be equivalent to a Symplectic Riccati System. This relationship is then used to obtain a discrete form of the above differential equation and both are linearisable.     

Solution of the Chazy system

We construct the solution of the Chazy system, which specifies the conditions for the existence of the Painlevé property for a third-order nonlinear equation with six poles.     

Heat equations and families of two-dimensional sigma functions

In the framework of S.P. Novikov’s program for boosting the effectiveness of thetafunction formulas of finite-gap integration theory, a system of differential equations for the parameters of the sigma function in genus 2 is constructed. A counterpart of this system in genus 1 is equivalent to the Chazy equation. On the basis of the obtained results, a two-dimensional analog of the Frobenius-Stickelberger connection is defined and calculated.     

Exact solutions to the partially integrable Eckhaus equation

A partially integrable extension of the Eckhaus equation is first converted to one real fourth-order equation. The only integrable case is isolated by simply solving a diophantine equation, and its linearizing transformation, not obvious at first glance, is shown to be the singular part transformation of Painlevé analysis. In the partially integrable case, three exact solutions are found by the truncation procedure. The third oneis a six-parameter solution, whose dependence on x is elliptic and dependence ont involves the equation of Chazy.Service de physique de l'état condensé, Centre d'études de Saclay, F-91190 Gif-sur-Yvette Cedex, France. Dienst Theoretische Natuurkunde, Vrije Universiteit Brussel, B-1050 Brussels, Belgium. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 99, No. 2, pp. 226–233, May, 1994.     

Egorov Hydrodynamic Chains, the Chazy Equation, and SL(2,ℂ)

The general solution of the system of differential equations describing Egorov hydrodynamic chains is constructed. The solution is given in terms of the elliptic sigma function. Invariants of the sigma function are expressed as differential polynomials in a solution of the Chazy equation. The orbits of the induced action of SL(2,) and degenerating operators in the space of solutions are described.     

Chazy Classes IX–XI Of Third-Order Differential Equations

In this article, we study Classes IX–XI of the 13 classes introduced by Chazy (1911) in his classification of third-order differential equations in the polynomial class having the Painlevé property. Classes IX and X are the only Chazy classes that have remained unsolved to this day, and they have been at the top of our "most wanted" list for some time. (There is an incorrect claim in the literature that these classes are unstable.) Here we construct their solutions in terms of hyperelliptic functions of genus 2, which are globally meromorphic. (We also add a parameter to Chazy Class X, overlooked in Chazy's original paper.) The method involves transforming to a more tractable class of fourth- and fifth-order differential equations, which is the subject of an accompanying paper (paper I). Most of the latter equations involve hyperelliptic functions and/or higher-order Painlevé transcendents. In the case of Chazy Class XI, the solution is elementary and well known, but there are interesting open problems associated with its coefficient functions, including the appearance of one of the aforementioned transcendents. In an appendix, we present the full list of Chazy equations (in the third-order polynomial class) and the solutions of those that are not dealt with in the body of this article.     

The differential equation satisfied by a plane curve of degree n

Eliminating the arbitrary coefficients in the equation of a generic plane curve of order n by computing sufficiently many derivatives, one obtains a differential equation. This is a projective invariant. The first one, corresponding to conics, has been obtained by Monge. Sylvester, Halphen, Cartan used invariants of higher order. The expression of these invariants is rather complicated, but becomes much simpler when interpreted in terms of symmetric functions.     

Rate‐independent processes in viscous solids at small strains

The so‐called generalized standard solids (of Halphen–Nguyen type) involving also activated typically rate‐independent processes such as plasticity, damage, or phase transformations are described as a system of a momentum equilibrium equation and a variational inequality for inelastic evolution of internal‐parameter variables. Various definitions of solutions are examined, especially from the viewpoint of the ability to combine rate‐independent processes and other rate‐dependent phenomena, as viscosity or also inertia. If those rate‐dependent phenomena are suppressed, then the system becomes fully rate‐independent. Illustrative examples are presented as well. Copyright © 2008 John Wiley & Sons, Ltd.     

Parameterizations of the Chazy Equation

The Chazy equation y?= 2yy″? 3y′² is derived from the automorphic properties of Schwarz triangle functions S(α, β, γ; z) . It is shown that solutions y which are analytic in the fundamental domain of these triangle functions, only correspond to certain values of α, β, γ . The solutions are then systematically constructed. These analytic solutions provide all known and one new parameterization of the Eisenstein series P, Q, R introduced by Ramanujan in his modular theories of signature 2, 3, 4, and 6.     

Equivalence of Third-Order Ordinary Differential Equations to Chazy Equations I–XIII

In the paper we solve the equivalence problem of the third-order ordinary differential equations quadratic in the second-order derivative. For this class of equations the invariants of the group of point equivalence transformations and the invariant differentiation operators are constructed. Using these results the invariants of 13 Chazy equations were calculated. We provide examples of finding equivalent equations by use of their invariants. Also two new examples of the equations linearizable by a local transformation are found. These are a particular case of Chazy–XII equation and a Schwarzian equation.     

A generalized Fitzhugh–Nagumo equation

In this paper we investigate mappings of the classical Fitzhugh–Nagumo equation to a generalized Fitzhugh–Nagumo equation. These mappings are invertible and transform the solutions of the classical Fitzhugh–Nagumo equation into solutions of the generalized Fitzhugh–Nagumo equation considered here. These mappings are found by considering the Lie point symmetries admitted by the classical Fitzhugh–Nagumo equation and the generalized Fitzhugh–Nagumo equation considered here. A particular example of a generalized Fitzhugh–Nagumo equation that satisfies the boundary conditions of the classical Fitzhugh–Nagumo equation is considered. Numerical solutions of the generalized Fitzhugh–Nagumo equation that do not satisfy the boundary conditions of the classical Fitzhugh–Nagumo equation are obtained by implementing the Method of Lines.     

On a generalization of the Chazy equation with a movable singular line

We generalize a third-order Chazy equation with a movable singular line, which has only negative resonances. For differential equations of order 2n+1 with resonances −1,−2, …, −(2n + 1), we study the convergence of the series representing their solutions, the existence of rational solutions, the invariance of these equations under certain transformations, and the existence of three-parameter solutions with a movable singular line.     

On the Solution of a System of Hamilton–Jacobi Equations of Special Form

The paper is concerned with the investigation of a system of first-order Hamilton–Jacobi equations. We consider a strongly coupled hierarchical system: the first equation is independent of the second, and the Hamiltonian of the second equation depends on the gradient of the solution of the first equation. The system can be solved sequentially. The solution of the first equation is understood in the sense of the theory of minimax (viscosity) solutions and can be obtained with the help of the Lax–Hopf formula. The substitution of the solution of the first equation in the second Hamilton–Jacobi equation results in a Hamilton–Jacobi equation with discontinuous Hamiltonian. This equation is solved with the use of the idea of M-solutions proposed by A. I. Subbotin, and the solution is chosen from the class of multivalued mappings. Thus, the solution of the original system of Hamilton–Jacobi equations is the direct product of a single-valued and multivalued mappings, which satisfy the first and second equations in the minimax and M-solution sense, respectively. In the case when the solution of the first equation is nondifferentiable only along one Rankine–Hugoniot line, existence and uniqueness theorems are proved. A representative formula for the solution of the system is obtained in terms of Cauchy characteristics. The properties of the solution and their dependence on the parameters of the problem are investigated.     

Almost biased mappings and almost compatible mappings are equivalent under some condition

The concepts of almost biased mappings and almost compatible mappings are introduced. Some variants of these concepts are also discussed. A common fixed point theorem for a pair of almost I-biased mappings in metric spaces is proved under certain condition. In the sequel, it is shown that almost biased mappings and almost compatible mappings are equivalent under some condition.     

Integrable Duffing’s maps and solutions of the Duffing equation

In the numerical integration of nonlinear differential equations, discretization of the nonlinear terms poses extra ambiguity in reducing the differential equation to a discrete difference equation. As for the cubic nonlinear Schrodinger equation, it was well known that there exists the corresponding discrete soliton system. Here, representing the discrete systems by the mappings, we explore structure of the integrable mappings. We introduce the first kind and the second kind of Duffing’s map, and investigate temporal evolution of the orbits. Although the smooth periodic orbits are in accord with the solutions of the Duffing equation, the integrable Duffing’s maps provide much wider variety of orbits.