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Giuseppe Gaeta

Publications and source records attributed to Giuseppe Gaeta.

51 records · Page 3Linked to original sources
arXiv2015

Speed selection for coupled wave equations

We discuss models for coupled wave equations describing interacting fields, focusing on the speed of travelling wave solutions. In particular, we propose a general mechanism for selecting and tuning the speed of the corresponding (multi-component) travelling wave solutions under certain physical conditions. A number of physical models (molecular chains, coupled Josephson junctions, propagation of kinks in chains of adsorbed atoms and domain walls) are considered as examples.

nlin.PS↗
arXiv2014

Simple and collective twisted symmetries

After the introduction of $λ$-symmetries by Muriel and Romero, several other types of so called "twisted symmetries" have been considered in the literature (their name refers to the fact they are defined through a deformation of the familiar prolongation operation); they are as useful as standard symmetries for what concerns symmetry reduction of ODEs or determination of special (invariant) solutions for PDEs and have thus attracted attention. The geometrical relation of twisted symmetries to standard ones has already been noted: for some type of twisted symmetries (in particular, $λ$ and $μ$-symmetries), this amounts to a certain kind of gauge transformation. In a previous review paper [G. Gaeta, "Twisted symmetries of differential equations", {\it J. Nonlin. Math. Phys.}, {\bf 16-S} (2009), 107-136] we have surveyed the first part of the developments of this theory; in the present paper we review recent developments. In particular, we provide a unifying geometrical description of the different types of twisted symmetries; this is based on the classical Frobenius reduction applied to distribution generated by Lie-point (local) symmetries.

math-ph↗
arXiv2014

Symmetry and Lie-Frobenius reduction of differential equations

Twisted symmetries, widely studied in the last decade, proved to be as effective as standard ones in the analysis and reduction of nonlinear equations. We explain this effectiveness in terms of a Lie-Frobenius reduction; this requires to focus not just on the prolonged (symmetry) vector fields but on the distributions spanned by these and on systems of vector fields in involution in Frobenius sense, not necessarily spanning a Lie algebra.

math-ph↗
arXiv2013

Dynamical systems and σ-symmetries

A deformation of the standard prolongation operation, defined on sets of vector fields in involution rather than on single ones, was recently introduced and christened "σ-prolongation"; correspondingly one has "σ-symmetries" of differential equations. These can be used to reduce the equations under study, but the general reduction procedure under σ-symmetries fails for equations of order one. In this note we discuss how σ-symmetries can be used to reduce dynamical systems, i.e. sets of first order ODEs in the form dx^a/dt = f^a (x).

math-ph↗
arXiv2012

A generalization of λ-symmetry reduction for systems of ODEs: σ-symmetries

We consider a deformation of the prolongation operation, defined on sets of vector fields and involving a mutual interaction in the definition of prolonged ones. This maintains the "invariants by differentiation" property, and can hence be used to reduce ODEs satisfying suitable invariance conditions in a fully algorithmic way, similarly to what happens for standard prolongations and symmetries.

math-ph↗
arXiv2012

Orbital reducibility and a generalization of lambda symmetries

We review the notion of reducibility and we introduce and discuss the notion of orbital reducibility for autonomous ordinary differential equations of first order. The relation between (orbital) reducibility and (orbital) symmetry is investigated and employed to construct (orbitally) reducible systems. By standard identifications, the notions extend to non-autonomous ODEs of first and higher order. Moreover we thus obtain a generalization of the lambda symmetries of Muriel and Romero. Several examples are given.

math-ph↗
arXiv2008

Twist solitons in complex macromolecules: from DNA to polyethylene

DNA torsion dynamics is essential in the transcription process; simple models for it have been proposed by several authors, in particular Yakushevich (Y model). These are strongly related to models of DNA separation dynamics such as the one first proposed by Peyrard and Bishop (and developed by Dauxois, Barbi, Cocco and Monasson among others), but support topological solitons. We recently developed a ``composite'' version of the Y model, in which the sugar-phosphate group and the base are described by separate degrees of freedom. This at the same time fits experimental data better than the simple Y model, and shows dynamical phenomena, which are of interest beyond DNA dynamics. Of particular relevance are the mechanism for selecting the speed of solitons by tuning the physical parameters of the non linear medium and the hierarchal separation of the relevant degrees of freedom in ``master'' and ``slave''. These mechanisms apply not only do DNA, but also to more general macromolecules, as we show concretely by considering polyethylene.

q-bio.BM↗
arXiv2007

Weierstrass's criterion and compact solitary waves

Weierstrass's theory is a standard qualitative tool for single degree of freedom equations, used in classical mechanics and in many textbooks. In this Brief Report we show how a simple generalization of this tool makes it possible to identify some differential equations for which compact and even semicompact traveling solitary waves exist. In the framework of continuum mechanics, these differential equations correspond to bulk shear waves for a special class of constitutive laws.

physics.class-ph↗
arXiv2004

Symmetry of stochastic equations

Symmetry methods are by now recognized as one of the main tools to attack deterministic differential equations (both ODEs and PDEs); the situation is quite different for what concerns stochastic differential equations: here, symmetry considerations are of course quite widely used by theoretical physicists, but a rigorous and general theory comparable to the one developed for deterministic equation is still lacking. In the following I will report on some work I have done on symmetries of stochastic (Ito) equations, and how these compare with the symmetries of the associated diffusion (Fokker-Planck) equations.

math-ph↗
arXiv2003

The Poincare'-Nekhoroshev map

We study a generalization of the familiar Poincaré map, first implicitely introduced by N.N. Nekhoroshev in his study of persistence of invariant tori in hamiltonian systems, and discuss some of its properties and applications. In particular, we apply it to study persistence and bifurcation of invariant tori.

math-ph↗
arXiv2002

Poincare' and Lie renormalized forms for regular singular points of vector fields in the plane

We discuss the local behaviour of vector fields in the plane $\R^2$ around a regular singular point, using recently introduced reduced normal forms, i.e. Poincaré and Lie renormalized forms [{\it Lett. Math. Phys.} {\bf 42} (1997), 103-114; {\it Ann. Inst. H. Poincaré (Phys. Theo.)} {\bf 70} (1999), 461-514; {\it Lett. Math. Phys.} {\bf 57} (2001), 41-60]. We give a complete classification, and provide explicit formulas, using Poincaré renormalized forms for non-degenerate cases, and Lie ones for certain degenerate cases. Both schemes are completely algorithmic, prove to be easy to implement, and only require to solve linear equations.

math-ph↗
arXiv2002

On persistence of invariant tori and a theorem by Nekhoroshev

We give a proof of a theorem by N.N. Nekhoroshev concerning Hamiltonian systems with $n$ degrees of freedom and $s$ integrals of motion in involution, where $1 \le s \le n$. Such a theorem ensures persistence of $s$-dimensional invariant tori under suitable nondegeneracy conditions generalizing Poincaré's condition on the Floquet multipliers.

math-ph↗
arXiv2001

Poincare' normal forms and simple compact Lie groups

We classify the possible behaviour of Poincaré-Dulac normal forms for dynamical systems in $R^n$ with nonvanishing linear part and which are equivariant under (the fundamental representation of) all the simple compact Lie algebras and thus the corresponding simple compact Lie groups. The ``renormalized forms'' (in the sense of previous work by the author) of these systems is also discussed; in this way we are able to simplify the classification and moreover to analyze systems with zero linear part. We also briefly discuss the convergence of the normalizing transformations.

math-ph↗
arXiv2001

Resonant normal forms as constrained linear systems

We show that a nonlinear dynamical system in Poincare'-Dulac normal form (in $\R^n$) can be seen as a constrained linear system; the constraints are given by the resonance conditions satisfied by the spectrum of (the linear part of) the system and identify a naturally invariant manifold for the flow of the ``parent'' linear system. The parent system is finite dimensional if the spectrum satisfies only a finite number of resonance conditions, as implied e.g. by the Poincare' condition. In this case our result can be used to integrate resonant normal forms, and sheds light on the geometry behind the classical integration method of Horn, Lyapounov and Dulac.

math-ph↗