Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · math-ph/0108004

Group foliation and non-invariant solutions of the heavenly equation

L. Martina·M. B. Sheftel·P. Winternitz
View original source ↗Read source document ↗Export metadata ↓

Abstract

The main physical result of this paper are exact analytical solutions of the heavenly equation, of importance in the general theory of relativity. These solutions are not invariant under any subgroup of the symmetry group of the equation. The main mathematical result is a new method of obtaining noninvariant solutions of partial differential equations with infinite dimensional symmetry groups. The method involves the compatibility of the given equations with a differential constraint, which is automorphic under a specific symmetry subgroup, the latter acting transitively on the submanifold of the common solutions. By studying the integrability of the resulting conditions, one can provide an explicit foliation of the entire solution manifold of the considered equations.

Explore related subjects

math-phmath.GRmath.MPnlin.SI

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

L. Martina, M. B. Sheftel, P. Winternitz. 2001-08-15. Group foliation and non-invariant solutions of the heavenly equation. https://doi.org/10.1088/0305-4470%2F34%2F43%2F310

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Massless Representations in Conformal Space and Their de Sitter Restrictions

The monograph offers a coherent and self-contained treatment of massless (ladder) representations of the conformal group U(2,2) and their restriction to the de Sitter group Sp(2,2), combining rigorous representation-theoretic analysis with fully explicit constructions. It systematically develops these representations, including the derivation of invariant bilinear forms and Casimir operators, and constructs vertex operators and two-point functions for low-helicity fields. A central and distinctive contribution is the introduction of a canonical Clifford-split-octonion framework, in which 8-component Majorana spinors are realized within an alternative composition algebra, providing a unified and intrinsically defined setting for the algebraic, spinorial, and geometric structures underlying the theory. By bridging abstract symmetry principles with concrete computational methods and physically motivated applications in quantum field theory and cosmology, the monograph advances both conceptual clarity and technical control. While primarily addressed to researchers in mathematical physics and related fields, the exposition is carefully structured to guide advanced graduate students through subtle constructions, maintaining accessibility without compromising mathematical precision.

math-ph↗
arXiv2026

Gauge invariant generalizations of the Proca equation and the Yang-Mills-Proca equation

The Proca equations (1936) are used in quantum field theory to describe vector bosons (spin 1) with a nonzero mass. The Proca equations are not gauge invariant. In contrast to Stueckelberg's approach (1938), this article presents a gauge invariant generalization of the Proca equation by introducing an additional vector field into the Proca equation. The results are extended to the Yang-Mills-Proca equations, leading to equations with non-Abelian gauge symmetry.

math-ph↗
arXiv2026

A Wigner-Lindblad phase space model for open quantum dynamics

We investigate the collisional quantum dynamics of an open system in contact with a thermal bath. We propose a particle-bath interaction model based on the Weyl quantization principle and the Wigner phase space description of quantum evolution. By factorizing the reduced Hamiltonian of the open system via supersymmetric quantum mechanics techniques, we construct system-dependent creation and annihilation operators for arbitrary potential landscapes. This allows us to extend the classical Fokker--Planck relaxation mechanism to incorporate Lindblad quantum stochastic jumps within the Wigner phase-space formalism. Through a semiclassical expansion of the Moyal star product to second order, we obtain a generalized Wigner--Lindblad kinetic model featuring position-dependent friction and diffusion terms. We establish the mathematical well-posedness of the Lindblad problem and we validate our approach through numerical simulations. Our model resolves unphysical long-time artifacts of standard quantum Fokker--Planck models in quartic double-well traps, accurately captures dissipative quantum state localization in 2D molecular potentials (4D phase space), and effectively evaluates electron transport and trapping times in InAs/InP resonant tunneling diodes.

math-ph↗