Global solutions of the biconfluent Heun equation

An algorithm is proposed for obtaining global solutions of the biconfluent Heun equation, which appears when dealing with a variety of physical problems. The procedure, which provides algebraic expressions of the solutions in the form of convergent series or asymptotic expansions, lies on the determination of the connection factors relating the solutions about the regular singular point at the origin and the irregular one at infinity. The algorithm is illustrated by examples.     

The 192 solutions of the Heun equation

A machine-generated list of local solutions of the Heun equation is given. They are analogous to Kummer's  solutions of the Gauss hypergeometric equation, since the two equations are canonical Fuchsian differential equations on the Riemann sphere with four and three singular points, respectively. Tabulation is facilitated by the identification of the automorphism group of the equation with  singular points as the Coxeter group  . Each of the expressions is labeled by an element of  . Of the ,  are equivalent expressions for the local Heun function  , and it is shown that the resulting order- group of transformations of  is isomorphic to the symmetric group . The isomorphism encodes each transformation as a permutation of an abstract four-element set, not identical to the set of singular points.

     

Degeneration of Heun equation solutions under fusion of singularities

Limiting transitions are considered in the process of the degeneration of the Heun equations. We show that eigenfunctions of the Heun equation are mapped into eigenfunctions of the degenerate Heun equation. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 110, No. 2, pp. 228–232, February, 1997.     

An algorithm to obtain global solutions of the double confluent Heun equation

A procedure is proposed to construct solutions of the double confluent Heun equation with a determinate behaviour at the singular points. The connection factors are expressed as quotients of Wronskians of the involved solutions. Asymptotic expansions are used in the computation of those Wronskians. The feasibility of the method is shown in an example, namely, the Schrödinger equation with a quasi-exactly-solvable potential.     

On reducing the Heun equation to the hypergeometric equation

The reductions of the Heun equation to the hypergeometric equation by polynomial transformations of its independent variable are enumerated and classified. Heun-to-hypergeometric reductions are similar to classical hypergeometric identities, but the conditions for the existence of a reduction involve features of the Heun equation that the hypergeometric equation does not possess; namely, its cross-ratio and accessory parameters. The reductions include quadratic and cubic transformations, which may be performed only if the singular points of the Heun equation form a harmonic or an equianharmonic quadruple, respectively; and several higher-degree transformations. This result corrects and extends a theorem in a previous paper, which found only the quadratic transformations. (SIAM J. Math. Anal. 10 (3) (1979) 655).     

Group algebras acting on the space of solutions of a special double confluent Heun equation

Theoretical and Mathematical Physics - We study properties of the space $$boldsymbol{Omega}$$ of solutions of a special double confluent Heun equation closely related to the model of a overdamped...     

Euler integral symmetries for a deformed Heun equation and symmetries of the Painlevé PVI equation

Euler integral transformations relate solutions of ordinary linear differential equations and generate integral representations of the solutions in a number of cases or relations between solutions of constrained equations (Euler symmetries) in some other cases. These relations lead to the corresponding symmetries of the monodromy matrices. We discuss Euler symmetries in the case of the simplest Fuchsian system that is equivalent to a deformed Heun equation, which is in turn related to the Painlevé PVI equation. The existence of integral symmetries of the deformed Heun equation leads to the corresponding symmetries of the PVI equation. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 155, No. 2, pp. 252–264, May, 2008.     

Series solutions of confluent Heun equations in terms of incomplete Gamma-functions

We present a simple systematic algorithm for construction of expansions of the solutions of ordinary differential equations with rational coefficients in terms of mathematical functions having indefinite integral representation. The approach employs an auxiliary equation involving only the derivatives of a solution of the equation under consideration. Using power-series expansions of the solutions of this auxiliary equation, we construct several expansions of the four confluent Heun equations'' solutions in terms of the incomplete Gamma-functions. In the cases of single- and double-confluent Heun equations the coefficients of the expansions obey four-term recurrence relations, while for the bi- and tri-confluent Heun equations the recurrence relations in general involve five terms. Other expansions for which the expansion coefficients obey recurrence relations involving more terms are also possible. The particular cases when these relations reduce to ones involving less number of terms are identified. The conditions for deriving closed-form finite-sum solutions via right-hand side termination of the constructed series are discussed.     

The asymptotic behavior of the discrete spectrum generated by the radial confluent Heun equation with close singularities

Against the background of one of the authors’ (S. Yu. Slavyanov’s) reminiscences of A. A. Bolibrukh, the asymptotic behavior of the spectral curves generated by the boundary-value problem for the confluent Heun equation (that is, an equation with two regular and one irregular singularity) is considered. The spectral curves are constructed for small values of one parameter (the distance between the regular singular points) depending on another parameter, which has the meaning of total charge in physics. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 38, Suzdal Conference-2004, Part 3, 2006.     

On the convergence of solutions of the enskog equation to solutions of the boltzmann equation

For the Enskog equation with a symmetrized kernel in a box an existence theorem is proved for initial data with finite mass, energy and entropy. Then by letting the diameter of the molecules go to zero we prove the weak convergence of solutions of the Enskog equation to solutions of the Boltzmann equation.     

Symmetries of the Confluent Heun Equation

The group of discrete symmetries of the confluent Heun equation is under consideration. Elementary and integral symmetries are generators of this group. Such symmetries give rise to respective symmetries of the connection matrix, which describes relations between solutions determined at different regular singular points. The Laplace integral symmetry relates the Stokes parameters and the connection matrix. Bibliography: 14 titles.     

Entire solutions of the KPP equation

This paper deals with the solutions defined for all time of the KPP equation u_t = u_xx + f(u),   0 < u(x,t) < 1, (x,t) ∈ ℝ², where ƒ is a KPP‐type nonlinearity defined in [0,1]: ƒ(0) = ƒ(1) = 0, ƒ′(0) > 0, ƒ′(1) < 0, ƒ > 0 in (0,1), and ƒ′(s) ≤ ƒ′(0) in [0,1]. This equation admits infinitely many traveling‐wave‐type solutions, increasing or decreasing in x. It also admits solutions that depend only on t. In this paper, we build four other manifolds of solutions: One is 5‐dimensional, one is 4‐dimensional, and two are 3‐dimensional. Some of these new solutions are obtained by considering two traveling waves that come from both sides of the real axis and mix. Furthermore, the traveling‐wave solutions are on the boundary of these four manifolds. © 1999 John Wiley & Sons, Inc.     

Algebraic-geometric solutions of the Krichever-Novikov equation

A zero-curvature representation with constant poles on an elliptic curve is obtained for the Krichever-Novikov equation. Algebraic-geometric solutions of this equation are constructed. The consideration is based on reducing the theta function of a two-sheet covering of an elliptic curve to the Prym theta functions of codimension one. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 121, No. 3, pp. 367–373, December, 1999.     

Stationary solutions of the vlasov-fokker-planck equation

We study stationary solutions of the Vlasov-Fokker-Planck Equation, a modification of the Vlasov-Poisson Equation, obtained by adding a diffusion term with respect to velocity. From a physical point of view it describes a plasma in thermal equilibrium. We prove existence of stationary solutions; for the mollified equation even an existence- and uniqueness theorem holds for sufficiently high temperature.