arXiv subjects

Mark E. Fels

Publications and source records attributed to Mark E. Fels.

8 recordsLinked to original sources

arXiv2026

On the Darboux Integrability of the Constant Mean Curvature One Equation for Surfaces in Hyperbolic 3 space

We show, using group theoretic constructions, that the system of elliptic partial differential equations whose solutions determine the constant mean curvature immersions into hyperbolic 3 space are equivalent to the elliptic Liouville equation. In particular, the Darboux integrability of each equation shows that each of these equations admit an equivalent quotient representation which descends to an equivalence of the equations. The explicit map providing the equivalence is given.

math.DG↗

arXiv2024

The Geometry of Darboux Integrable Elliptic Systems

We characterize real elliptic differential systems whose solutions can be expressed in terms of holomorphic solutions to an associated holomorphic Pfaffian system $\mathcal H$ on a complex manifold. In particular, these elliptic systems arise as quotients by a group $G$ of the real differential system generated by the real and imaginary parts of $\mathcal H$, such that $G$ is the real form of a complex Lie group $K$ which is a symmetry group of $\mathcal H$. Subject to some mild genericity assumptions, we show that such elliptic systems are characterized by a property known as Darboux integrability. Examples discussed include first- and second-order elliptic PDE and PDE systems in the plane.

math.DG↗

arXiv2019

Variational Operators, Symplectic Operators, and the Cohomology of Scalar Evolution Equations

For a scalar evolution equation $u_t=K(t,x,u,u_x,\ldots, u_n), n\geq 2$ the cohomology spaces $H^{1,s}({\mathcal R}^\infty)$ vanishes for $s\geq 3$ while the space $H^{1,2}({\mathcal R}^\infty)$ is isomorphic to the space of variational operators. The cohomology space $H^{1,2}({\mathcal R}^\infty)$ is also shown to be isomorphic to the space of symplectic operators for $u_t=K$ for which the equation is Hamiltonian. Third order scalar evolution equations admitting a first order symplectic (or variational) operator are characterized. The symplectic nature of the potential form of a bi-Hamiltonian evolution equation is also presented.

math.DG↗

arXiv2015

On the Construction of Simply Connected Solvable Lie Groups

Let $ω_\mathfrak{g}$ be a Lie algebra valued differential $1$-form on a manifold $M$ satisfying the structure equations $d ω_\mathfrak{g} + \frac{1}{2} ω_\mathfrak{g}\wedge ω_\mathfrak{g}=0$ where $\mathfrak{g}$ is solvable. We show that the problem of finding a smooth map $ρ:M\to G$, where $G$ is an $n$-dimensional solvable Lie group with Lie algebra $\mathfrak{g}$ and left invariant Maurer-Cartan form $τ$, such that $ρ^* τ= ω_\mathfrak{g}$ can be solved by quadratures and the matrix exponential. In the process we give a closed form formula for the vector fields in Lie's third theorem for solvable Lie algebras. A further application produces the multiplication map for a simply connected $n$-dimensional solvable Lie group using only the matrix exponential and $n$ quadratures. Applications to finding first integrals for completely integrable Pfaffian systems with solvable symmetry algebras are also given.

math.DG↗

arXiv2015

Backlund Transformations for Darboux Integrable Differential Systems: Examples and Applications

In the article arXiv:1108.5443 we established a general group-theoretical approach to the construction of Bäcklund transformations. We then showed how this construction can be applied to construct Bäcklund transformation between equations which are Darboux integrable. Here we give a number of detailed examples and new applications which demonstrate the theory. In particular our final example demonstrates how our group theoretical approach produces all the Bäcklund transformations in arXiv:0707.4408. We also prove, using group methods, a non-existence theorem for Bäcklund transformatons which disagrees with part 2 of Theorem 1 in arXiv:0707.4408.

math.DG↗

arXiv2014

Backlund Transformations for Darboux Integrable Differential Systems

We give a new mechanism for constructing Backlund transformations by using symmetry reduction of differential systems. We then characterize a family of Backlund transformations between Darboux integrable systems where the Backlund transformation can be constructed by the proposed symmetry reduction method.

math.DG↗

arXiv2013

The Cauchy Problem for Darboux Integrable Systems and Non-Linear d'Alembert Formulas

To every Darboux integrable system there is an associated Lie group $G$ which is a fundamental invariant of the system and which we call the Vessiot group. This article shows that solving the Cauchy problem for a Darboux integrable partial differential equation can be reduced to solving an equation of Lie type for the Vessiot group $G$. If the Vessiot group $G$ is solvable then the Cauchy problem can be solved by quadratures. This allows us to give explicit integral formulas, similar to the well known d'Alembert's formula for the wave equation, to the initial value problem with generic non-characteristic initial data.

math.DG↗

arXiv2001

The Principle of Symmetric Criticality in General Relativity

We consider a version of Palais' Principle of Symmetric Criticality (PSC) that is applicable to the Lie symmetry reduction of Lagrangian field theories. PSC asserts that, given a group action, for any group-invariant Lagrangian the equations obtained by restriction of Euler-Lagrange equations to group-invariant fields are equivalent to the Euler-Lagrange equations of a canonically defined, symmetry-reduced Lagrangian. We investigate the validity of PSC for local gravitational theories built from a metric. It is shown that there are two independent conditions which must be satisfied for PSC to be valid. One of these conditions, obtained previously in the context of transverse symmetry group actions, provides a generalization of the well-known unimodularity condition that arises in spatially homogeneous cosmological models. The other condition seems to be new. The conditions that determine the validity of PSC are equivalent to pointwise conditions on the group action alone. These results are illustrated with a variety of examples from general relativity. It is straightforward to generalize all of our results to any relativistic field theory.

gr-qc↗