Painleve Equation Research Articles

Reducing Modified Formulas for Sinh-Gordon Equation to the Painleve Equations

Reducing Modified Formulas for Sinh-Gordon Equation to the Painleve Equations

Asymptotics of the determinant of the modified Bessel functions and the second Painlevé equation

In the paper, we consider the extended Gross–Witten–Wadia unitary matrix model by introducing a logarithmic term in the potential. The partition function of the model can be expressed equivalently in terms of the Toeplitz determinant with the [Formula: see text]-entry being the modified Bessel functions of order [Formula: see text], [Formula: see text]. When the degree [Formula: see text] is finite, we show that the Toeplitz determinant is described by the isomonodromy [Formula: see text]-function of the Painlevé III equation. As a double scaling limit, we establish an asymptotic approximation of the logarithmic derivative of the Toeplitz determinant, expressed in terms of the Hastings–McLeod solution of the inhomogeneous Painlevé II equation with parameter [Formula: see text]. The asymptotics of the leading coefficient and recurrence coefficient of the associated orthogonal polynomials are also derived. We obtain the results by applying the Deift–Zhou nonlinear steepest descent method to the Riemann–Hilbert problem for orthogonal polynomials on the Hankel loop. The main concern here is the construction of a local parametrix at the critical point [Formula: see text], where the [Formula: see text]-function of the Jimbo–Miwa Lax pair for the inhomogeneous Painlevé II equation is involved.

Applying a Power Transformation to the Orbit of the 2nd Painleve Equation and Solving Differential Equations with Polynomial Right-hand Sides Via the 2nd Painleve Transcendent and in Polynomials

The 2nd Painleve equation is considered as a representative of the second-order class of ordinary differential equations (ODEs) with polynomial right-hand sides, as well as of the more general second-order class of equations with fractional polynomial right-hand sides. The second Painlevé equation with three terms on the right side has an orbit in the class of fractional polynomial equations with respect to the pseudogroup of the 36th order, and in the absence of the 3rd term – the 60th order. This paper presents a power transformation with an arbitrary parameter that preserves the polynomial or fractional polynomial form of the equations. This power-law transformation is applied to the orbital equations of the 2nd Painlevé equation with three and two terms on the right-hand sides of the equations. Pseudogroups of transformations induced by the above-mentioned pseudogroups of the 36th and 60th orders are constructed. All equations with one-constant arbitrariness corresponding to the vertices of the graphs of induced pseudogroups are found. General solutions of all found equations are obtained through the 2nd Painlevé transcendental or in polynomials. A theorem is presented that allows, using the scaling operation, to find general solutions to all the above equations with arbitrary coefficients.

Reduction of quad-equations consistent around a cuboctahedron I: Additive case

In this paper, we consider a reduction of a new system of partial difference equations, which was obtained in our previous paper (Joshi and Nakazono, arXiv:1906.06650) and shown to be consistent around a cuboctahedron. We show that this system reduces to $A_2^{(1)\ast}$-type discrete Painleve equations by considering a periodic reduction of a three-dimensional lattice constructed from overlapping cuboctahedra.

Global Meromorphy of Solutions of the Painlevé Equations and Their Hierarchies

We show that all local holomorphic solutions of all equations constituting the hierarchies of the first and second Painleve equations can be analytically continued to meromorphic functions on the whole complex plane. We also present a new conceptual proof of the fact that all local holomorphic solutions of the first, second, and fourth Painleve equations are globally meromorphic.

First-Order Ode Systems Generating Confluent Heun Equations

We study the relation between linear second-order equations that are confluent Heun equations, namely, the biconfluent and triconfluent Heun equations, and first-order linear systems of equations generating Painleve equations. The generation process is interpreted in physical terms as antiquantization. Technically, the study in volves manipulations with polynomials. The complexity of computations sometimes requires using computer algebra systems. Bibliography: 13 titles.

Riccati Equations Revisited: Linearization and Analytic Interpretation of Instanton-Type Solutions

As a prototype of Painleve equations, we discuss instanton-type solutions of the Riccati equation with a large parameter in this paper. Based on the linearization and its modified version, we provide an analytic interpretation of instanton-type solutions through the Borel resummation technique.

Analytic Properties of Solutions to Equations in the Generalized Hierarchy of the Second Painlevé Equation

We consider the analytic properties of solutions to equations of arbitrary order in the generalized hierarchy of the second Painleve equation. The local properties of solutions, the Backlund transformations, and rational solutions and their representations via generalized Yablonskii–Vorob’ev polynomials are studied.

Discrete Painlev 'e Equations

This expository article written for the Notices of the American Mathematical Society provides an overview of transcendental functions arising as solutions of the discrete Painleve equations, for which the developments of the last two decades have been rich and dynamic. These equations arise at the center of many viewpoints: random matrix theory, algebra, algebraic geometry, dynamical systems and the theory of transcendental functions. The purpose of this article is to reveal this confluence and modern perspectives on it.

Rational Solutions of Equations Associated with the Second Painlevé Equation

Nonlinear differential equations associated with the second Painleve equation are considered. Transformations for solutions of the singular manifold equation are presented. It is shown that rational solutions of the singular manifold equation are determined by means of the Yablonskii-Vorob’ev polynomials. It is demonstrated that rational solutions for some differential equations are also expressed via the Yablonskii-Vorob’ev polynomials.

Point equivalence of second-order ordinary differential equations to the fifth Painlevé equation with one and two nonzero parameters

We consider the problem of the equivalence of scalar second-order ordinary differential equations under invertible point transformations. To solve this problem in the case of Painleve equations, we use previously constructed invariants of a family of equations whose right-hand sides have a cubic nonlinearity in the first derivative. We obtain the conditions for point equivalence to the fifth Painleve equation if two or three of its parameters are equal to zero.

Existence of rational solutions for q-difference Painleve equations

This article studies properties of meromorphic solutions for several types of q-difference Painleve equations. We obtain conditions for the existence, and the form of rational solutions for two classes of q-difference Painleve equations. Also for a solution f we obtain results about the fixed points, the exponents of convergence of poles of . Our results extend previous theorems given in the references. For more information see https://ejde.math.txstate.edu/Volumes/2020/14/abstr.html

On the Convergence of Formal Exotic Series Solutions of an ODE

We propose a sufficient condition for the convergence of a complex power type formal series of the form $$\varphi =\sum _{k=1}^{\infty }\alpha _k(x^{\mathrm{i}\gamma })\,x^k$$, where $$\alpha _k$$ are functions meromorphic at the origin and $$\gamma \in {{\mathbb {R}}}\setminus \{0\}$$, that satisfies an analytic ordinary differential equation (ODE) of a general type. An example of such a type formal solutions of the third Painleve equation is presented and the proposed sufficient condition is applied to check their convergence; moreover, the accumulation of movable poles of these solutions near the critical point is discussed.

Discrete Painlev\xe9 equation, Miwa variables and string equation in 5d matrix models

The modern version of conformal matrix model (CMM) describes conformal blocks in the Dijkgraaf-Vafa phase. Therefore it possesses a determinant representation and becomes a Toda chain T-function only after a peculiar Fourier transform in internal dimensions. Moreover, in CMM Hirota equations arise in a peculiar discrete form (when the couplings of CMM are actually Miwa time-variables). Instead, this integrability property is actually independent of the measure in the original hypergeometric integral. To get hypergeometric functions, one needs to pick up a very special T-function, satisfying an additional “string equation”. Usually its role is played by the lowest L-1 Virasoro constraint, but, in the Miwa variables, it turns into a finite-difference equation with respect to the Miwa variables. One can get rid of these differences by rewriting the string equation in terms of some double ratios of the shifted T-functions, and then these ratios satisfy more sophisticated equations equivalent to the discrete Painleve equations by M. Jimbo and H. Sakai (q-PVI equation). They look much simpler in the q-deformed (“5d“) matrix model, while in the “continuous” limit q → 1 to 4d one should consider the Miwa variables with non-unit multiplicities, what finally converts the simple discrete Painleve q-PVI into sophisticated differential Painleve VI equations, which will be considered elsewhere.

Necessary Conditions for Point Equivalence of Second-Order Odes to the Sixth Painlevé Equation

We consider the equivalence problem for projective-type scalar second-order ordinary differential equations with respect to invertible point changes of variables. Invariants of the equivalence transformation group of this family of equations are used to find necessary conditions for equivalence to the sixth Painleve equation.

Asymptotic behaviour of the third Painlevé transcendents in the space of initial values

We study the asymptotic behavior of solutions of the fourth Painleve equation as the independent variable goes to infinity in its space of (complex) initial values, which is a generalization of phase space described by Okamoto. We show that the limit set of each solution is compact and connected and, moreover, that any solution that is not rational has an infinite number of poles and infinite number of zeros.

On Various Approaches to Asymptotics of Solutions to the Third Painlevé Equation in a Neighborhood of Infinity

We examine asymptotic expansions of the third Painleve transcendents for αδ ≠ = 0 and γ = 0 in the neighborhood of infinity in a sector of aperture <2π by the method of dominant balance). We compare intermediate results with results obtained by methods of three-dimensional power geometry. We find possible asymptotics in terms of elliptic functions, construct a power series, which represents an asymptotic expansion of the solution to the third Painleve equation in a certain sector, estimate the aperture of this sector, and obtain a recurrent relation for the coefficients of the series.

The Symmetric Semi-classical Orthogonal Polynomials of Class Two and Some of Their Extensions

We study a large class of orthogonal polynomials, containing the semi-classical symmetric orthogonal polynomials of class two. We show difference equations for the recurrence coefficients of the orthogonal polynomials as well as for related quantities. Some of these recurrences are identified with discrete Painleve equations.

On the Exact Gevrey Order of Formal Puiseux Series Solutions to the Third Painlevé Equation

In this paper, we study the third Painleve equation with parameters γ = 0, αδ ≠ 0. The Puiseux series formally satisfying this equation (after a certain change of variables) asymptotically approximate of Gevrey order one solutions to this equation in sectors with vertices at infinity. We present a family of values of the parameters δ = −β2/2 ≠ 0 such that these series are of exact Gevrey order one, and hence diverge. We prove the 1-summability of them and provide analytic functions which are approximated of Gevrey order one by these series in sectors with the vertices at infinity.

Quantum curves and q-deformed Painlevé equations

We propose that the grand canonical topological string partition functions satisfy finite-difference equations in the closed string moduli. In the case of genus one mirror curve, these are conjectured to be the q-difference Painleve equations as in Sakai’s classification. More precisely, we propose that the tau functions of q-Painleve equations are related to the grand canonical topological string partition functions on the corresponding geometry. In the toric cases, we use topological string/spectral theory duality to give a Fredholm determinant representation for the above tau functions in terms of the underlying quantum mirror curve. As a consequence, the zeroes of the tau functions compute the exact spectrum of the associated quantum integrable systems. We provide details of this construction for the local $$\mathbb {P}^1\times \mathbb {P}^1$$ case, which is related to q-difference Painleve with affine $$A_1$$ symmetry, to SU(2) Super Yang–Mills in five dimensions and to relativistic Toda system.