Local and nonlocal solvable structures in ODEs reduction

Abstract

Solvable structures, likewise solvable algebras of local symmetries, can be used to integrate scalar ODEs by quadratures. Solvable structures, however, are particularly suitable for the integration of ODEs with a lack of local symmetries. In fact, under regularity assumptions, any given ODE always admits solvable structures even though finding them in general could be a very difficult task. In practice a noteworthy simplification may come by computing solvable structures which are adapted to some admitted symmetry algebra. In this paper we consider solvable structures adapted to local and nonlocal symmetry algebras of any order (i.e., classical and higher). In particular we introduce the notion of nonlocal solvable structure.

References in corpus (3)

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• On the relation between standard and $\mu$-symmetries for PDEs
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Cited by in corpus (12)

• Symmetry of stochastic non-variational differential equations
• Nonlocal Symmetries, Telescopic Vector Fields and $\lambda$-Symmetries of Ordinary Differential Equations
• ${\mathcal{C}}^{\infty }$-symmetries of distributions and integrability
• A gauge-theoretic description of $\mu$-prolongations, and $\mu$-symmetries of differential equations
• Solvable structures for evolution PDEs admitting differential constraints
• Symmetry and Lie-Frobenius reduction of differential equations
• ${\mathcal{C}}^{\infty }$-structures in the integration of involutive distributions
• Nonlocal interpretation of $\lambda$-variational symmetry-reduction method
• On the Geometry of Twisted Symmetries: Gauging and Coverings
• Twisted symmetries and integrable systems
• $\lambda$-symmetries and Jacobi Last Multiplier
• Twisted symmetries of differential equations