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

Abstract

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.

Non-canonical extension of θ-functions and modular integrability of ϑ-constants

“…9.4. Good examples of Lie's symmetries application to Chazy's equation (42) and many other Jacobi's 'modular/elliptic' equations are presented in nice works [25] and [68]. As with solutions (33) and (55) we can write solutions to system (54) that respect Jacobi's identity (43).…”

“…Consequences. An interrelation between (74) and Jacobi's formula (68) needs to be understood if the elliptic curve (66) has already been given in the canonical form (67) as…”

“…For lack of space we omit the proofs of these statements, but they can be partially derived from the procedure of integration of the equations, which is detailed in § 9.4. Good examples of the application of Lie's symmetries to Chazy's equation (5.7) and many other of Jacobi's 'modular/elliptic' equations are presented in [25,74]. As with (4.7) and (6.13) we can write solutions to (6.12) that respect Jacobi's identity (6.1).…”

“…Elliptic integrals K and K ′ were introduced above. They give an inversion of the first formula in (98), that is an equivalent of Jacobi's formula (68):…”

Jacobi Last Multiplier and Lie Symmetries: A Novel Application of an Old Relationship

“…As far as we know, this is only the second appearance of special functions in the generators of Lie symmetries admitted by a nonlinear differential equation. The preceding example, which involves elliptic integrals, can be found in [55].…”