Integration of some examples of geodesic flows via solvable structures

Solvable structures are particularly useful in the integration by quadratures of ordinary differential equations. Nevertheless, for a given equation, it is not always possible to compute a solvable structure. In practice, the simplest solvable structures are those adapted to an already known system of symmetries. In this paper we propose a method of integration which uses solvable structures suitably adapted to both symmetries and first integrals. In the variational case, due to Noether theorem, this method is particularly effective as illustrated by some examples of integration of the geodesic flows

A class of third order partial differential equations describing spherical or pseudospherical surfaces

Third order equations, which describe spherical surfaces (ss) or pseudospherical surfaces (pss), of the form $$\nu\,z_{t}-\lambda\,z_{xxt}=A(z,z_{x},z_{xx})\,z_{xxx}+B(z,z_{x},z_{xx})$$ with $\nu$, $\lambda$ $\in$ $\mathbb{R}$, $\nu^2+\lambda^2\neq 0$, are considered. These equations are equivalent to the structure equations of a metric with Gaussian curvature $K=1$ or $K=-1$, respectively. Alternatively they can be seen as the compatibility condition of an associated $\mathfrak{su}(2)$-valued or $\mathfrak{sl}(2,\mathbb{R})$-valued linear problem, also referred to as a zero curvature representation. Under certain assumptions we obtain an explicit classification for equations of the considered form that describe ss or pss, in terms of some arbitrary differentiable functions. Several examples of such equations, which describe also a number of already known equations, are provided by suitably choosing the arbitrary functions

Symmetries and first integrals for non-variational equations

For a class of exterior ideals, we present a method associating first integrals of the characteristic distributions to symmetries of the ideal. The method is applied, under some assumptions, to the study of first integrals of ordinary differential equations and first order partial differential equations as well as to the determination of first integrals for integrable distributions of vector fields

Partecipante di Progetto giovani ricercatori del GNFM "Simmetrie e riduzione per PDE, principi di sovrapposizione e strutture nonlocali"

Coordinatrice del progetto è stata Paola Morando. Lo scopo del progetto è stato quello di approfondire lo studio delle tecniche di riduzione e integrazione di sistemi differenziali esterni. Particolare attenzione è stata rivolta all'uso di simmetrie, anche nonlocali. Parte dei risultati ottenuti è stato oggetto di pubblicazione su riviste scientifiche internazionali. I risultati più recenti, soprattutto quelli relativi allo studio di equazioni integrabili secondo Darboux e principi di sovrapposizione nonlineare (anche per sistemi di PDE nonlineari) sarà oggetto di future pubblicazioni

Contact symmetries of the elliptic Euler-Darboux equation

We study contact symmetries of an elliptic equation parametrizing some Ricci flat metrics with bidimensional Killing orbits.The variational nature of such symmetries is investigated as well