Plancherel-Rotach asymptotics for certain basic hypergeometric series

In this work we study the chaotic and periodic asymptotics for the confluent basic hypergeometric series. For a fixed q∈(0,1), the asymptotics for Euler's q-exponential, q-Gamma function Γq(x), q-Airy function of K. Kajiwara, T. Masuda, M. Noumi, Y. Ohta and Y. Yamada, Ramanujan function (q-Airy function), Jackson's q-Bessel function of second kind, Ismail-Masson orthogonal polynomials (q⁻¹-Hermite polynomials), Stieltjes-Wigert polynomials, q-Laguerre polynomials could be derived as special cases.     

On some aspects of Bäcklund transformations

The Painlevé differential equations (P₂-P₆) possess Bäcklund transformations which relate one solution to another solution either of the same equation, with different values of the parameters, or another such equation. We review a method for deriving difference equations, the discrete Painlevé equations in particular, from Bäcklund transformations of the continuous Painlevé equations. Then, we prove the existence of an algebraic formula relating three inconsecutive solutions of the same Bäcklund hierarchy for P₃ and P₄.     

Value distribution of meromorphic solutions of certain difference Painlevé equations

The Borel exceptional value and the exponents of convergence of poles, zeros and fixed points of finite order transcendental meromorphic solutions for difference Painlevé I and II equations are estimated. And the forms of rational solutions of the difference Painlevé II equation and the autonomous difference Painlevé I equation are also given. It is also proved that the non-autonomous difference Painlevé I equation has no rational solution.     

Folding transformations of the Painlevé equations

New symmetries of the Painlevé differential equations, called folding transformations, are determined. These transformations are not birational but algebraic transformations of degree 2, 3, or 4. These are associated with quotients of the spaces of initial conditions of each Painlevé equation. We make the complete list of such transformations up to birational symmetries. We also discuss correspondences of special solutions of Painlevé equations.Acknowledgement The authors wish to thank Prof. Yosuke Ohyama, Prof. Shun Shimomura, and Dr. Yoshikatsu Sasaki for valuable discussions.     

Coulomb Gas Representation for Rational Solutions of the Painlevé Equations

We consider rational solutions for a number of dynamic systems of the type of the nonlinear Schrödinger equation, in particular, the Levi system. We derive the equations for the dynamics of poles and Bäcklund transformations for these solutions. We show that these solutions can be reduced to rational solutions of the Painlevé IV equation, with the equations for the pole dynamics becoming the stationary equations for the two-dimensional Coulomb gas in a parabolic potential. The corresponding Coulomb systems are derived for the Painlevé II–VI equations. Using the Hamiltonian formalism, we construct the spin representation of the Painlevé equations.     

Invariant Simultaneous Solutions of Evolution Equations of Integrable Hierarchies

We construct classical point symmetry groups for joint pairs of evolution equations (systems of equations) of integrable hierarchies related to the auxiliary equation of the method of the inverse problem of second order. For the two cases: the hierarchy of Korteweg--de Vries (KdV) equations and of the systems of Kaup equations, we construct simultaneous solutions invariant with respect to the symmetry group. The problem of the construction of these solutions can be reduced, respectively, to the first and second Painlevé equations depending on a parameter. The Painlevé equations are supplemented by the linear evolution equations defining the deformation of the solution of the corresponding Painlevé equation.     

Discrete Symmetries of Systems of Isomonodromic Deformations of Second-Order Fuchsian Differential Equations

We compute the discrete affine group of Schlesinger transformations for isomonodromic deformations of a Fuchsian system of second-order differential equations. These transformations are treated as isomorphisms between the moduli spaces of logarithmic sl(2)-connections with given eigenvalues of the residues on ¹. The discrete structure is computed with the use of the modification technique for bundles with connections. The result generalizes the well-known classical computations of symmetries of the hypergeometric equation, the Heun equation, and the sixth Painlevé equation.     

Integral representation of solutions to Fuchsian system and Heun's equation

We obtain integral representations of solutions to special cases of the Fuchsian system of differential equations and Heun's differential equation. In particular, we calculate the monodromy of solutions to the Fuchsian equation that corresponds to Picard's solution of the sixth Painlevé equation, and to Heun's equation.     

Solutions of Discrete-Velocity Boltzmann Equations via Bateman and Riccati Equations

We propose several approaches for solving two discrete-velocity Boltzmann equations using the rescaling ansatz and the truncated Painlevé expansions. We use solutions of the two- and three-dimensional Bateman equations for the singularity manifold conditions to reduce the problem to Riccati equations. Both equations fail the Painlevé test.     

On non-isospectral flows,Painlevé equations,and symmetries of differential and difference equations

We identify the Painlevé Lax pairs with those corresponding to stationary solutions of non-isospectral flows, both for partial differential equations and differential-difference equations. We discuss symmetry reductions of integrable differential-difference equations and show that, in contrast with the continuous case, where Painlevé equations naturally arise, in the discrete case the so-called discrete Painlevé equations cannot be obtained in this way. Actually, symmetry reductions of integrable differential-difference equations naturally provide delay Painlevé equations.In Memory of Prof. M. C. PolivanovDipartimento di Fisica, P. le A. Moro 2, 00185 Roma, Istituto Nazionale di Fisica Nucleare, Sezione di Roma, Italy. Departamento de Fisica Teorica, Facultad de Fisicas, Universidad Complutense, 28040 Madrid, Spain. W561@emducm11.bitnet. Published in Teoreticheskaya i Matematicheskaya Fizika, Vol. 93, No. 3, pp. 473–480, December, 1992.     

Amalgamations of the Painlevé Equations

We present new hierarchies of nonlinear ordinary differential equations (ODEs) that are generalizations of the Painlevé equations. These hierarchies contain the Painlevé equations as special cases. We emphasize the sixth-order ODEs. Special solutions for one of them are expressed via the general solutions of the P ₁ and P ₂ equations and special cases of the P ₃ and P ₅ equations. Four of the six Painlevé equations can be considered special cases of these sixth-order ODEs. We give linear representations for solving the Cauchy problems for the hierarchy equations using the inverse monodromy transform.     

Solitary wave solutions to some classes of nonlinear evolution type equations using inverse variational methods

Invariants of reduced forms of a p.d.e. are obtainable from a variational principle even though the p.d.e. itself does not admit a Lagrangian. The reductions carry all the advantages regarding Noether symmetries and double reductions via first integrals or conserved quantities. The examples we consider are nonlinear evolution type equations like the general form of the Fizhugh–Nagumo and KdV–Burgers equations. Some aspects of Painlevé properties of the reduced equations are also obtained.     

Full symmetry groups, Painlevé integrability and exact solutions of the nonisospectral BKP equation

Based on the generalized symmetry group method presented by Lou and Ma [Lou and Ma, Non-Lie symmetry groups of (2 + 1)-dimensional nonlinear systems obtained from a simple direct method, J. Phys. A: Math. Gen. 38 (2005) L129], firstly, both the Lie point groups and the full symmetry group of the nonisospectral BKP equation are obtained, at the same time, a relationship is constructed between the new solutions and the old ones of equation. Secondly, the nonisospectral BKP can be proved to be Painlevé integrability by combining the standard WTC approach with the Kruskal’s simplification, some solutions are obtained by using the standard truncated Painlevé expansion. Finally, based on the relationship by the generalized symmetry group method and some solutions by using the standard truncated Painlevé expansion, some interesting solution are constructed.     

Tropical representation of Weyl groups associated with certain rational varieties

Starting from certain rational varieties blown-up from N(P¹), we construct a tropical, i.e., subtraction-free birational, representation of Weyl groups as a group of pseudo-isomorphisms of the varieties. We develop an algebro-geometric framework of τ-functions as defining functions of exceptional divisors on the varieties. In the case where the corresponding root system is of affine type, our construction yields a class of (higher order) q-difference Painlevé equations and its algebraic degree grows quadratically.     

Universal characters, integrable chains and the Painlevé equations

The universal character is a generalization of the Schur polynomial attached to a pair of partitions; see (Adv. Math. 74 (1989) 57). We prove that the universal character solves the Darboux chain. The N-periodic closing of the chain is equivalent to the Painlevé equation of type . Consequently we obtain an expression of rational solutions of the Painlevé equations in terms of the universal characters.     

The discrete Korteweg-de Vries equation

We review the different aspects of integrable discretizations in space and time of the Korteweg-de Vries equation, including Miura transformations to related integrable difference equations, connections to integrable mappings, similarity reductions and discrete versions of Painlevé equations as well as connections to Volterra systems.     

Lie point symmetries, partial Noether operators and first integrals of the Painlevé-Gambier equations

We utilize the Lie-Tressé linearization method to obtain linearizing point transformations of certain autonomous nonlinear second-order ordinary differential equations contained in the Painlevé-Gambier classification. These point transformations are constructed using the Lie point symmetry generators admitted by the underlying Painlevé-Gambier equations. It is also shown that those Painlevé-Gambier equations which have a few Lie point symmetries and hence are not linearizable by this method can be integrated by a quadrature. Moreover, by making use of the partial Lagrangian approach we obtain time dependent and time independent first integrals for these Painlevé-Gambier equations which have not been reported in the earlier literature. A comparison of the results obtained in this paper is made with the ones obtained using the generalized Sundman linearization method.     

Analytical solutions to Fisher’s equation with time-variable coefficients

This paper provides analytical solutions to the generalized Fisher equation with a class of time varying diffusion coefficients. To accomplish this we use the Painlevé property for partial differential equations as defined by Weiss in 1983 in “The Painlevé property for partial-differential equations”. This was first done for the variable coefficient Fisher’s equation by Ö?ün and Kart in 2007; we build on this work, finding additional solutions with a weaker restriction on the trial solution. We also use the same technique to find solutions to Fisher’s equation with time-dependent coefficients for both diffusion and nonlinear terms. Lastly we compute specific solutions to illustrate their behaviors.     

On tronquée solutions of the first Painlevé hierarchy

It is well known that, due to Boutroux, the first Painlevé equation admits solutions characterized by divergent asymptotic expansions near infinity in specified sectors of the complex plane. In this paper, we show that such solutions exist for higher order analogues of the first Painlevé equation (the first Painlevé hierarchy) as well.