27 research outputs found
Superposition rules for higher-order systems and their applications
Superposition rules form a class of functions that describe general solutions
of systems of first-order ordinary differential equations in terms of generic
families of particular solutions and certain constants. In this work we extend
this notion and other related ones to systems of higher-order differential
equations and analyse their properties. Several results concerning the
existence of various types of superposition rules for higher-order systems are
proved and illustrated with examples extracted from the physics and mathematics
literature. In particular, two new superposition rules for second- and
third-order Kummer--Schwarz equations are derived.Comment: (v2) 33 pages, some typos corrected, added some references and minor
commentarie
Lie families: theory and applications
We analyze families of non-autonomous systems of first-order ordinary
differential equations admitting a common time-dependent superposition rule,
i.e., a time-dependent map expressing any solution of each of these systems in
terms of a generic set of particular solutions of the system and some
constants. We next study relations of these families, called Lie families, with
the theory of Lie and quasi-Lie systems and apply our theory to provide common
time-dependent superposition rules for certain Lie families.Comment: 23 pages, revised version to appear in J. Phys. A: Math. Theo
Anti-self-dual Riemannian metrics without Killing vectors, can they be realized on K3?
Explicit Riemannian metrics with Euclidean signature and anti-self dual
curvature that do not admit any Killing vectors are presented. The metric and
the Riemann curvature scalars are homogenous functions of degree zero in a
single real potential and its derivatives. The solution for the potential is a
sum of exponential functions which suggests that for the choice of a suitable
domain of coordinates and parameters it can be the metric on a compact
manifold. Then, by the theorem of Hitchin, it could be a class of metrics on
, or on surfaces whose universal covering is .Comment: Misprints in eqs.(9-11) corrected. Submitted to Classical and Quantum
Gravit
Partner symmetries of the complex Monge-Ampere equation yield hyper-Kahler metrics without continuous symmetries
We extend the Mason-Newman Lax pair for the elliptic complex Monge-Amp\`ere
equation so that this equation itself emerges as an algebraic consequence. We
regard the function in the extended Lax equations as a complex potential. We
identify the real and imaginary parts of the potential, which we call partner
symmetries, with the translational and dilatational symmetry characteristics
respectively. Then we choose the dilatational symmetry characteristic as the
new unknown replacing the K\"ahler potential which directly leads to a Legendre
transformation and to a set of linear equations satisfied by a single real
potential. This enables us to construct non-invariant solutions of the Legendre
transform of the complex Monge-Amp\`ere equation and obtain hyper-K\"ahler
metrics with anti-self-dual Riemann curvature 2-form that admit no Killing
vectors.Comment: submitted to J. Phys.
Discrete analogues of the Liouville equation
The notion of Laplace invariants is transferred to the lattices and discrete
equations which are difference analogs of hyperbolic PDE's with two independent
variables. The sequence of Laplace invariants satisfy the discrete analog of
twodimensional Toda lattice. The terminating of this sequence by zeroes is
proved to be the necessary condition for existence of the integrals of the
equation under consideration. The formulae are presented for the higher
symmetries of the equations possessing integrals. The general theory is
illustrated by examples of difference analogs of Liouville equation.Comment: LaTeX, 15 pages, submitted to Teor. i Mat. Fi
Integrability of Lie systems through Riccati equations
Integrability conditions for Lie systems are related to reduction or
transformation processes. We here analyse a geometric method to construct
integrability conditions for Riccati equations following these approaches. This
approach provides us with a unified geometrical viewpoint that allows us to
analyse some previous works on the topic and explain new properties. Moreover,
this new approach can be straightforwardly generalised to describe
integrability conditions for any Lie system. Finally, we show the usefulness of
our treatment in order to study the problem of the linearisability of Riccati
equations.Comment: Corrected typo