528,067 research outputs found
Ordinary differential equations described by their Lie symmetry algebra
The theory of Lie remarkable equations, i.e. differential equations
characterized by their Lie point symmetries, is reviewed and applied to
ordinary differential equations. In particular, we consider some relevant Lie
algebras of vector fields on and characterize Lie remarkable
equations admitted by the considered Lie algebras.Comment: 17 page
The th Hilbert problem on algebraic limit cycles
For real planar polynomial differential systems there appeared a simple
version of the th Hilbert problem on algebraic limit cycles: {\it Is there
an upper bound on the number of algebraic limit cycles of all polynomial vector
fields of degree ?} In [J. Differential Equations, 248(2010), 1401--1409]
Llibre, Ram\'irez and Sadovskia solved the problem, providing an exact upper
bound, in the case of invariant algebraic curves generic for the vector fields,
and they posed the following conjecture: {\it Is the maximal
number of algebraic limit cycles that a polynomial vector field of degree
can have?}
In this paper we will prove this conjecture for planar polynomial vector
fields having only nodal invariant algebraic curves. This result includes the
Llibre {\it et al}\,'s as a special one. For the polynomial vector fields
having only non--dicritical invariant algebraic curves we answer the simple
version of the 16th Hilbert problem.Comment: 16. Journal Differential Equations, 201
Modular differential equations for characters of RCFT
We discuss methods, based on the theory of vector-valued modular forms, to
determine all modular differential equations satisfied by the conformal
characters of RCFT; these modular equations are related to the null vector
relations of the operator algebra. Besides describing effective algorithmic
procedures, we illustrate our methods on an explicit example.Comment: 13 page
Topologies of continuity for Carathéodory delay differential equations with applications in non-autonomous dynamics
Producción CientÃficaWe study some already introduced and some new strong and weak topologies of integral type to provide continuous dependence on continuous initial data for the solutions of non-autonomous Carathéodory delay differential equations. As a consequence, we obtain new families of continuous skew-product semiflows generated by delay differential equations whose vector fields belong to such metric topological vector spaces of Lipschitz Carathéodory functions. Sufficient conditions for the equivalence of all or some of the considered strong or weak topologies are also given. Finally, we also provide results of continuous dependence of the solutions as well as of continuity of the skew-product semiflows generated by Carathéodory delay differential equations when the considered phase space is a Sobolev space.MINECO/FEDER MTM2015-66330-PH2020-MSCA-ITN-2014 643073 CRITICS
Invariant Modules and the Reduction of Nonlinear Partial Differential Equations to Dynamical Systems
We completely characterize all nonlinear partial differential equations
leaving a given finite-dimensional vector space of analytic functions
invariant. Existence of an invariant subspace leads to a re duction of the
associated dynamical partial differential equations to a system of ordinary
differential equations, and provide a nonlinear counterpart to quasi-exactly
solvable quantum Hamiltonians. These results rely on a useful extension of the
classical Wronskian determinant condition for linear independence of functions.
In addition, new approaches to the characterization o f the annihilating
differential operators for spaces of analytic functions are presented.Comment: 28 pages. To appear in Advances in Mathematic
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