All Binomial-Type Painlevé Equations of the Second Order and Degree Three or Higher

In this paper we construct all rational Painlevé-type differential equations which take the binomial form, (d²y/dx²)ⁿ = F(x,y,dy/dx), where n ≥ 3, the case n = 2 having previously been treated in Cosgrove and Scoufis [1]. While F is assumed to be rational in the complex variables y and y′ and locally analytic in x, it is shown that the Painlevé property together with the absence of intermediate powers of y″ forces F to be a polynomial in y and y′. In addition to the six classes of second-degree equations found in the aforementioned paper, we find nine classes of higher-degree binomial Painlevé equations, denoted BP-VII,..., BP-XV, of which the first seven are new. Two of these equations are of the third degree, two of the fourth degree, three of the sixth degree, and two of arbitrary degree n. All equations are solved in terms of the first, second or fourth Painlevé transcendents, elliptic functions, or quadratures. In the appendices, we discuss certain closely related classes of second-order nth equations (not necessarily of Painlevé type) which can also be solved in terms of Painlevé transcendents or elliptic functions.     

Painlevé Classification of All Semilinear Partial Differential Equations of the Second Order. II. Parabolic and Higher Dimensional Equations

This paper extends the work of the previous paper (I) on the Painlevé classification of second-order semilinear partial differential equations to the case of parabolic equations in two independent variables, u_xx = F(x, y, u, u_x, u_y), and irreducible equations in three or more independent variables of the form, Σ_ijR_ij (x₁,…, x_n)u,_ij = F(x₁,…, x_n; u,₁,…, u,_n). In each case, F is assumed to be rational in u and its first derivatives and no other simplifying assumptions are made. In addition to the 22 hyperbolic equations found in paper I, we find 10 equivalence classes of parabolic equations with the Painlevé property, denoted PS-I, PS-I1,…, PS-X, equation PS-II being a generalization of Burgers' equation denoted the Forsyth-Burgers equation, and 13 higher-dimensional Painlevé equations, denoted GS-I, GS-II,…, GS-XIII. The lists are complete up to the equivalence relation of Möbius transformations in u and arbitrary changes of the independent variables. In order to avoid repetition, the proofs are sketched very briefly in cases where they closely resemble those for the corresponding hyperbolic problem. Every equation is solved by transforming to a linear partial differential equation, from which it follows that there are no non trivial soliton equations among the two classes of Painlevé equations treated in this paper.     

Whitney umbrella and slow-motion bifurcations of relaxation-type equations

Smooth orbital normal forms of generic one-parametric families of slow motion of relaxation-type equations with two-dimensional slow variable are obtained near a singular point of the type Whitney-fold of the equation folding when the slow velocity is not tangent to the set of critical values of the folding. For example, a generic family for a generic value of the parameter is described by the germ at the origin of either the equations (dy/dx)² = x(x - y)² or (dy/dx)² = x found by V. I. Arnold and M. Cibrario, respectively, after an appropriate choice of smooth local coordinates fibered over the parameter spaces.Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 7, Suzdal Conference-1, 2003.This revised version was published online in April 2005 with a corrected cover date.     

Nonlinear Transition Layers—The Second Painleve Transcendent

We investigate a large class of weakly nonlinear second-order ordinary differential equations with slowly varying coefficients. We show that the standard two-timing perturbation solution is not valid during the transition from oscillatory to exponentially decaying behavior. In all cases this difficulty is remedied by a nonlinear transition layer, whose leading-order character is described by one special nonlinear differential equation known as the second Painlevé transcendent (in essence a nonlinear Airy equation). The method of matched asymptotic expansions yields the desired connection formula. The second Painlevé transcendent also provides two other types of transitions: (1) between weakly nonlinear solutions (either oscillatory or exponentially decaying) and special fully nonlinear solutions, and (2) between two of these special nonlinear solutions. These special solutions are of three: different kinds: (a) slowly varying stable equilibrium solutions, (b) “exploding” solutions, and (c) solutions depending on both the fast and slow scales (which emerge from the unstable zero equilibrium solution).     

The two-dimensional eikonal equation

We study the two-dimensional eikonal equation ψ _x ² + ψ _y ² = 1/v ²(x, y). We carry out the group analysis of the equation, establish a connection between the group properties and geometric characteristics of the Riemannian space with the metric ds ² = [dx ² + dy ²]/v ²(x, y). We select the most important classes of equations and derive some conditions for reducibility of a given equation to an equation of one of those classes. We find a condition for two equations to be equivalent (the theorem of seven invariants). For the equations corresponding to Riemannian spaces of constant curvature, we obtain explicit formulas for the solutions describing the wave front for a point source and also the ray equations.     

Generalization of the simplest equation method for nonlinear non-autonomous differential equations

It is known that the simplest equation method is applied for finding exact solutions of autonomous nonlinear differential equations. In this paper we extend this method for finding exact solutions of non-autonomous nonlinear differential equations (DEs). We applied the generalized approach to look for exact special solutions of three Painlevé equations. As ODE of lower order than Painlevé equations the Riccati equation is taken. The obtained exact special solutions are expressed in terms of the special functions defined by linear ODEs of the second order.     

The 2×2 matrix Schlesinger system and the Belavin-Polyakov-Zamolodchikov system

We show that the Belavin-Polyakov-Zamolodchikov equation of the minimal model of conformal field theory with the central charge c = 1 for the Virasoro algebra is contained in a system of linear equations that generates the Schlesinger system with 2×2 tmatrices. This generalizes Suleimanov’s result on the Painlevé equations. We consider the properties of the solutions, which are expressible in terms of the Riemann theta function.     

On the relation between the continuous and discrete Painlevé equations

A method for deriving difference equations (the discrete Painlevé equations in particular) from the Bäcklund transformations of the continuous Painlevé equations is discussed. This technique can be used to derive several of the known discrete painlevé equations (in particular, the first and second discrete Painlevé equations and some of their alternative versions). The Painlevé equations possess hierarchies of rational solutions and one-parameter families of solutions expressible in terms of the classical special functions for special values of the parameters. Hence, the aforementioned relations can be used to generate hierarchies of exact solutions for the associated discrete Painlevé equations. Exact solutions of the Painlevé equations simultaneously satisfy both a differential equation and a difference equation, analogously to the special functions.     

Ill-posedness and local ill-posedness concepts in hilbert spaces *

In this paper, we study ill-posedness concepts of nonlinear and linear operator equations in a Hilbert space setting. Such ill-posedness information may help to select appropriate optimization approaches for the stable approximate solution of inverse problems, which are formulated by the operator equations. We define local ill-posedness of a nonlinear operator equation F(x) = y ₀ in a solution point x ₀:and consider the interplay between the nonlinear problem and its linearization using the Fréchet derivative F′(x ₀). To find a corresponding ill-posedness concept for the linearized equation we define intrinsic ill-posedness for linear operator equations A x = y and compare this approach with the ill-posedness definitions due to Hadamard and Nashed     

Euler integral symmetries for the confluent Heun equation and symmetries of the Painlevé equation PV

Euler integral symmetries relate solutions of ordinary linear differential equations and generate integral representations of the solutions in several cases or relations between solutions of constrained equations. These relations lead to the corresponding symmetries of the monodromy matrices for the differential equations. We discuss Euler symmetries in the case of the deformed confluent Heun equation, which is in turn related to the Painlevé equation PV. The existence of symmetries of the linear equations leads to the corresponding symmetries of the Painlevé equation of the Okamoto type. The choice of the system of linear equations that reduces to the deformed confluent Heun equation is the starting point for the constructions. The basic technical problem is to choose the bijective relation between the system parameters and the parameters of the deformed confluent Heun equation. The solution of this problem is quite large, and we use the algebraic computing system Maple for this.     

Integral transformation of solutions for a Fuchsian-class equation corresponding to the Okamoto transformation of the Painlevé VI equation

We show that under the Euler integral transformation with the kernel (x−z)^−α, some solutions of the Fuchs equations (the original pair for the Painlevé VI equation) pass into solutions of a system of the same form with the parameters changed according to the Okamoto transformation. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 3, pp. 355–364, March, 2006.     

Operator-theoretic solution of stochastic systems

A (stochastic) operator-theoretic approach leads to expresssions for inverses of linear and nonlinear stochastic operators—useful for the solution of linear or nonlinear stochastic differential equations. Operator equations are developed for inverses of linear or nonlinear stochastic operators. Series expressions are obtained which allow writing the solution y=$\text{F}$^?1x of the operator equation $\text{F}$y=x. Special cases are studied in which $\text{F}$ may be linear or nonlinear, deterministic or stochastic in various combinations.     

Second-degree Painlevé equations and their contiguity relations

We study second-order, second-degree systems related to the Painlevé equations which possess one and two parameters. In every case we show that by introducing a quantity related to the canonical Hamiltonian variables it is possible to derive such a second-degree equation. We investigate also the contiguity relations of the solutions of these higher-degree equations. In most cases these relations have the form of correspondences, which would make them non-integrable in general. However, as we show, in our case these contiguity relations are indeed integrable mappings, with a single ambiguity in their evolution (due to the sign of a square root).     

Some Algebraic Approach for the Second Painlevé Equation Using the Optimal Homotopy Asymptotic Method (OHAM)

The study of Painlevé equations has increased during the last years, due to the awareness that these equations and their solutions can accomplish good results both in the field of pure mathematics and in theoretical physics. In this paper we introduced the optimal homotopy asymptotic method (OHAM) approach to propose analytic approximate solutions to the second Painlevé equation. The advantage of this method is that it provides a simple algebraic expression that can be used for further developments while maintaining good performance and fitting closely the numerical solution.     

Almost continuability of solutions of differential equations

We introduce the notion of almost continuability of the solution of the differential equation of first order dy/dx = f(x, y) to the whole real axis. We give a criterion for the almost continuability of solutions for the case in which the right-hand side of the equation is a meromorphic function of one variable y: f(x, y) = g(y). As an example, we work out the case of a rational and, in particular, an entire function g(y).     

Power expansions for the self-similar solutions of the modified Sawada-Kotera equation

The fourth-order ordinary differential equation that defines the self-similar solutions of the Kaup—Kupershmidt and Sawada—Kotera equations is studied. This equation belongs to the class of fourth-order analogues of the Painlevé equations. All the power and non-power asymptotic forms and expansions near points z = 0, z = ∞ and near an arbitrary point z = z ₀ are found by means of power geometry methods. The exponential additions to the solutions of the studied equation are also determined.      

Quicksilver Solutions of a q‐Difference First Painlevé Equation

In this paper, we present new, unstable solutions, which we call quicksilver solutions, of a q‐difference Painlevé equation in the limit as the independent variable approaches infinity. The specific equation we consider in this paper is a discrete version of the first Painlevé equation (qP_I), whose phase space (space of initial values) is a rational surface of type . We describe four families of almost stationary behaviors, but focus on the most complicated case, which is the vanishing solution. We derive this solution's formal power series expansion, describe the growth of its coefficients, and show that, while the series is divergent, there exist true analytic solutions asymptotic to such a series in a certain q‐domain. The method, while demonstrated for qP_I, is also applicable to other q‐difference Painlevé equations.     

The Generating Function for the Airy Point Process and a System of Coupled Painlevé II Equations

For a wide class of Hermitian random matrices, the limit distribution of the eigenvalues close to the largest one is governed by the Airy point process. In such ensembles, the limit distribution of the k th largest eigenvalue is given in terms of the Airy kernel Fredholm determinant or in terms of Tracy–Widom formulas involving solutions of the Painlevé II equation. Limit distributions for quantities involving two or more near‐extreme eigenvalues, such as the gap between the k th and the ℓth largest eigenvalue or the sum of the k largest eigenvalues, can be expressed in terms of Fredholm determinants of an Airy kernel with several discontinuities. We establish simple Tracy–Widom type expressions for these Fredholm determinants, which involve solutions to systems of coupled Painlevé II equations, and we investigate the asymptotic behavior of these solutions.     

Asymptotic formulas for n-diameters of certain compacta in L2[0, 1]

In the current article the order of the Kolmogorov n-diameters of compacta, determined by the operatorsLy =p (x)dy/dx +q (x)y, Ly = [–d²/dx² +q (x) d/dx]^r y in L₂[0, 1] with a bound on the order of the error is studied and asymptotic formulas for d_n as a function of p(x), q(x), and r are derived.Translated from Matematicheskie Zametki, Vol. 20, No. 3, pp. 331–340, September, 1976.