Related papers: Automorphic Schwarzian equations
In this paper, we investigate the non-modular solutions to the Schwarz differential equation $\{f,\tau \}=sE_4(\tau)$ where $E_4(\tau)$ is the weight 4 Eisenstein series and $s$ is a complex parameter. In particular, we provide explicit…
In this paper we study the modular differential equation $y''+s\,E_4\, y=0$ where $E_4$ is the weight 4 Eisenstein series and $s=\pi^2r^2$ with $r=n/m$ being a rational number in reduced form such that $m\geq 7$. This study is carried out…
For every positive integer $r$, we solve the modular Schwarzian differential equation $\{h,\tau\}=2\pi^2r^2E_4$, where $E_4$ is the weight 4 Eisenstein series, by means of equivariant functions on the upper half-plane. This paper…
In this paper, we explore the modular differential equation $\displaystyle y'' + F(z)y = 0$ on the upper half-plane $\mathbb{H}$, where $F$ is a weight 4 modular form for $\Gamma_0(2)$. Our approach centers on solving the associated…
We study second-order modular differential equations whose solutions transform equivariantly under the modular group. In the reducible case, we construct all such solutions using an explicit ansatz involving Eisenstein series and the…
The Schwarzian derivative plays a fundamental role in complex analysis, differential equations, and modular forms. In this paper, we investigate its higher-order generalizations, known as higher Schwarzians, and their connections to…
A set of nonlinear differential equations associated with the Eisenstein series of the congruent subgroup $\Gamma_0(2)$ of the modular group $SL_2(\mathbb{Z})$ is constructed. These nonlinear equations are analogues of the well known…
The Schwarzian derivative is invariant under SL(2,R)-transformations and, as thus, any function of it can be used to determine the equation of motion or the Lagrangian density of a higher derivative SL(2,R)-invariant 1d mechanics or the…
The purpose of this paper is to solve various differential equations having Eisenstein series as coefficients using various tools and techniques. The solutions are given in terms of modular forms, modular functions and equivariant forms.
The Schwarzian equations satisfied by certain Hauptmoduls (i.e., uniformizing functions for Riemann surfaces of genus zero) are derived from the Picard-Fuchs equations for families of elliptic curves and associated surfaces. The…
The method of nonlinear realizations is used to clarify some conceptual and technical issues related to the Schwarzian mechanics. It is shown that the Schwarzian derivative arises naturally, if one applies the method to SL(2,R) times R…
In this paper, we consider the problem when a differential equation y"(z)=Q(z)y(z) is Fuchsian on H* and apparent on H, where Q(z) is a meromorphic modular form of weight 4 on SL(2,Z) and H denotes the complex upper half-plane. Such a…
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…
Sachs has derived quaternion field equations that fully exploit the underlying symmetry of the principle of general relativity, one in which the fundamental 10 component metric field is replaced by a 16 component four-vector quaternion.…
We provide a full and unbiased solution to the Dyson-Schwinger equation illustrated for $\phi^4$ theory in 2D. It is based on an exact treatment of the functional derivative $\partial \Gamma / \partial G$ of the 4-point vertex function…
In the present article we define the algebra of differential modular forms and we prove that it is generated by Eisenstein series of weight $2,4$ and 6. We define Hecke operators on them, find some analytic relations between these…
In this paper, we study the discrete fractional Schr\"{o}dinger equation $$ (-\Delta)^\alpha u+h(x) u=f(x,u),\quad x\in \mathbb{Z}^d,$$ where $d\in\mathbb{N}^*,\,\alpha \in(0, 1)$ and the nonlocal operator $(-\Delta)^\alpha $ is defined by…
A novel method for finding the eigenvalues of a Sturm-Liouville problem is developed. Following the minimalist approach the problem is transformed to a single first-order differential equation with appropriate boundary conditions. Although…
Physicists such as Green, Vanhove, et al show that differential equations involving automorphic forms govern the behavior of gravitons. One particular point of interest is solutions to $(\Delta-\lambda)u=E_{\alpha} E_{\beta}$ on an…
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…