Schwarzian Differential Equation Research Articles

All meromorphic solutions of the autonomous Schwarzian differential equations

AbstractThis paper considers a specific autonomous Schwarzian differential equation given by and presents a complete characterization of its transcendental meromorphic solutions, thus confirming a conjecture posed by Liao and Wu recently. By combining our results with those obtained in a previous paper (Math. Z. 2 (2022), 1657–1672), we are able to explicitly construct all transcendental meromorphic solutions of the autonomous Schwarzian differential equations. Additionally, we identify all nonconstant rational solutions of the autonomous Schwarzian differential equations, with the exception of one special case.

Equivariant solutions to modular Schwarzian equations

For every positive integer r, we solve the modular Schwarzian differential equation {h,τ}=2π2r2E4, where E4 is the weight 4 Eisenstein series, by means of equivariant functions on the upper half-plane. This paper supplements previous works by the authors [20,21], where the same equation has been solved for infinite families of rational values of r. This also leads to the solutions to the modular differential equation y″+r2π2E4y=0 for every positive integer r. These solutions are quasi-modular forms for SL2(Z) if r is even or for the subgroup of index 2, SL2(Z)2, if r is odd.

Exact meromorphic solutions of Schwarzian differential equations

This paper studies exact meromorphic solutions of the autonomous Schwarzian differential equations. All transcendental meromorphic solutions of five canonical types (among six) of the autonomous Schwarzian differential equations are constructed explicitly. In particular, the solutions of four types are shown to be elliptic functions. Also, all transcendental meromorphic solutions that are locally injective or possess a Picard exceptional value are characterized for the remaining canonical type.

Some functional transcendence results around the Schwarzian differential equation.

This paper centers around proving variants of the Ax-Lindemann-Weierstrass (ALW) theorem for analytic functions which satisfy Schwarzian differential equations. In previous work, the authors proved the ALW theorem for the uniformizers of genus zero Fuchsian groups, and in this work, we generalize that result in several ways using a variety of techniques from model theory, galois theory and geometry.

Uniqueness for meromorphic solutions of Schwarzian differential equation

<abstract><p>Let $ f $ be a meromorphic function, $ R $ be a nonconstant rational function and $ k $ be a positive integer. In this paper, we consider the Schwarzian differential equation of the form</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{align*} \left[\frac{f'''}{f'}-\frac{3}{2}\left(\frac{f''}{f'}\right)^{2}\right]^{k} = R(z). \end{align*} $\end{document} </tex-math></disp-formula></p> <p>We investigate the uniqueness of meromorphic solutions of the above Schwarzian differential equation if the meromorphic solution $ f $ shares three values with any other meromorphic function.</p></abstract>

Meromorphic solutions of difference equations originated from Schwarzian differential equation

Let f (z) be a meromorphic functions with finite order , R(z) be a nonconstant rational function and k be a positive integer. In this paper, we consider the difference equation originated from Schwarzian differential equation, which is of form [?3f(z)?f(z)- 3/2(?2|(z))2]k = R(z)(?f (z))2k. We investigate the uniqueness of meromorphic solution f of difference Schwarzian equation if f shares three values with any meromrphic function. The exact forms of meromorphic solutions f of difference Schwarzian equation are also presented.

New exact solutions of (3 + 1)‐dimensional generalized Kadomtsev‐Petviashvili equation using a combination of lie symmetry and singular manifold methods

A new combination of Lie symmetry and Singular Manifold methods has been employed to study (3 + 1)‐dimensional generalized Kadomtsev‐Petviashvili (KP). Infinite‐dimensional space of Lie vectors has been established. Single and dual linear combinations of Lie vectors are used after appropriate calculations of the arbitrary functions to reduce the equation to an ordinary differential equation (ODE). The resulting ODE is then analytically solved through the singular manifold method which resulted in a Bäcklund truncated series with seminal analysis leading to a Schwarzian differential equation in the Eigenfunction φ (η). Solving this differential equation leads to new analytical solutions.

Schwarzian conditions for linear differential operators with selected differential Galois groups

We show that non-linear Schwarzian differential equations emerging from covariance symmetry conditions imposed on linear differential operators with hypergeometric function solutions can be generalized to arbitrary order linear differential operators with polynomial coefficients having selected differential Galois groups. For order three and order four linear differential operators we show that this pullback invariance up to conjugation eventually reduces to symmetric powers of an underlying order-two operator. We give, precisely, the conditions to have modular correspondences solutions for such Schwarzian differential equations, which was an open question in a previous paper. We analyze in detail a pullbacked hypergeometric example generalizing modular forms, that ushers a pullback invariance up to operator homomorphisms. We finally consider the more general problem of the equivalence of two different order-four linear differential Calabi–Yau operators up to pullbacks and conjugation, and clarify the cases where they have the same Yukawa couplings.

Exact solutions of Calgero–Bogoyavlenskii–Schiff equation using the singular manifold method after Lie reductions

Calgero–Bogoyavlenskii–Schiff (CBS) equation is analytically solved through two successive reductions into an ordinary differential equation (ODE) through a set of optimal Lie vectors. During the second reduction step, CBS equation is reduced using hidden vectors. The resulting ODE is then analytically solved through the singular manifold method in three steps; First, a Bäcklund truncated series is obtained. Second, this series is inserted into the ODE, and finally, a seminal analysis leads to a Schwarzian differential equation in the eigenfunction φ(η). Solving this differential equation leads to new analytical solutions. Then, through two backward substitution steps, the original dependent variable is recovered. The obtained results are plotted for several Lie hidden vectors and compared with previous work on CBS equation using Lie transformations. Copyright © 2017 John Wiley & Sons, Ltd.

Schwarzian differential equations associated to Shimura curves of genus zero

Schwarzian differential equations associated to Shimura curves of genus zero

Schwarzian differential equations and Hecke eigenforms on Shimura curves

AbstractLet X be a Shimura curve of genus zero. In this paper, we first characterize the spaces of automorphic forms on X in terms of Schwarzian differential equations. We then devise a method to compute Hecke operators on these spaces. An interesting by-product of our analysis is the evaluation and other similar identities.

On the growth of meromorphic solutions of the Schwarzian differential equations

We study the properties of meromorphic solutions of the Schwarzian differential equations in the complex plane by using some techniques from the study of the class W p . We find some upper bounds of the order of meromorphic solutions for some types of the Schwarzian differential equations. We also show that there are no wandering domains nor Baker domains for meromorphic solutions of certain Schwarzian differential equations.

A Lie Symmetry Connection between Jacobi’s Modular Differential Equation and Schwarzian Differential Equation

In [18] Jacobi introduced a third-order nonlinear ordinary differential equation which links two different moduli of an elliptic integral. In the present paper Lie group analysis is applied to that equation named Jacobi’s modular differential equation. A six-dimensional Lie symmetry algebra is obtained and its symmetry generators are found to be given in terms of elliptic integrals. As a consequence the transformation between Jacobi’s modular differential equation and the well-known Schwarzian differential equation is derived.

Bers embedding of the Teichmüller space of a once-punctured torus

In this note, we present a method of computing monodromies of projective structures on a once-punctured torus. This leads to an algorithm numerically visualizing the shape of the Bers embedding of a one-dimensional Teichmüller space. As a by-product, the value of the accessory parameter of a four-times punctured sphere will be calculated in a numerical way as well as the generators of a Fuchsian group uniformizing it. Finally, we observe the relation between the Schwarzian differential equation and Heun’s differential equation in this special case.

On differential equations satisfied by modular forms

We use the theory of modular functions to give a new proof of a result of P. F. Stiller, which asserts that, if t is a non-constant meromorphic modular function of weight 0 and F is a meromorphic modular form of weight k with respect to a discrete subgroup of SL 2 (ℝ) commensurable with SL 2 (ℤ), then F, as a function of t, satisfies a (k+1)-st order linear differential equation with algebraic functions of t as coefficients. Furthermore, we show that the Schwarzian differential equation for the modular function t can be extracted from any given (k+1)-st order linear differential equation of this type. One advantage of our approach is that every coefficient in the differential equations can be relatively easily determined.

Arithmetic properties of mirror map and quantum coupling

We study some arithmetic properties of the mirror maps and the quantum Yukawa coupling for some 1-parameter deformations of Calabi-Yau manifolds. First we use the Schwarzian differential equation, which we derived previously, to characterize the mirror map in each case. For algebraic K3 surfaces, we solve the equation in terms of the $J$-function. By deriving explicit modular relations we prove that some K3 mirror maps are algebraic over the genus zero function field ${\bf Q}(J)$. This leads to a uniform proof that those mirror maps have integral Fourier coefficients. Regarding the maps as Riemann mappings, we prove that they are genus zero functions. By virtue of the Conway-Norton conjecture (proved by Borcherds using Frenkel-Lepowsky-Meurman's Moonshine module), we find that these maps are actually the reciprocals of the Thompson series for certain conjugacy classes in the Griess-Fischer group. This also gives, as an immediate consequence, a second proof that those mirror maps are integral. We thus conjecture a surprising connection between K3 mirror maps and the Thompson series. For threefolds, we construct a formal nonlinear ODE for the quantum coupling reduced $mod\ p$. Under the mirror hypothesis and an integrality assumption, we derive $mod~p$ congruences for the Fourier coefficients. For the quintics, we deduce (at least for $5\not{|}d$) that the degree $d$ instanton numbers $n_d$ are divisible by $5^3$ -- a fact first conjectured by Clemens.

Numerical conformal mapping of circular arc polygons

Howell, L.H., Numerical conformal mapping of circular arc polygons, Journal of Computational and Applied Mathematics 46 (1993) 7–28. Solutions to the Schwarzian differential equation are conformal maps from the upper half-plane to circular arc polygons, plane regions bounded by straight line segments and arbitrary arcs of circles. We develop methods for numerically integrating this equation, both directly and through the use of a related linear differential equation. Particular attention is given to the behavior near corner singularities. We also derive alternate versions of the transformation which map from the unit disk and from an infinite strip. While the former may be of primarily theoretical interest, the latter can be used to map highly elongated regions such as channels for internal flow problems. Such regions are difficult or impossible to map from the disk or the half-plane due to the so-called crowding phenomenon.

On the monodromy group for the super Schwarzian differential equation

The first variation of the monodromy group associated with a super Schwarzian differential equation is calculated. The relation between the monodromy period and the Fenchel–Nielsen deformation of a super Riemann surface is presented.

Admissible solutions of the Schwarzian differential equation

AbstractLet R(z, w) be a rational function of w with meromorphic coefficients. It is shown that if the Schwarzian equation possesses an admissible solution, then , where αj, are distinct complex constants. In particular, when R(z, w) is independent of z, it is shown that if (*) possesses an admissible solution w(z), then by some Möbius transformation u = (aw + b) / (cw + d) (ad – bc ≠ 0), the equation can be reduced to one of the following forms: where τj (j = 1, … 4) are distinct constants, and σj (j = 1, … 4) are constants, not necessarily distinct.

Painleve analysis of the two-dimensional Burgers equation

A Painleve analysis of the two-dimensional Burgers equation is carried out and used to obtain a restricted Backlund transformation that maps a subclass of the solutions of the 1+2D Burgers equation onto a linear heat-like equation. Alternatively, the Backlund transformation can be expressed as a map onto the derivative of the one-dimensional Burgers equation in appropriate dependent and independent variables. The singularity analysis also yields a further class of solutions obtained by solving a Schwarzian differential equation.