We expose a unified computational approach to integrable structures (including recursion, Hamiltonian, and symplectic operators) based on geometrical theory of partial differential equations.

In this paper we provide a translation of a paper by T. Levi-Civita, published in 1899, about the correspondence between symmetries and conservation laws for Hamilton’s equations. We discuss the results of this paper and their relationship with the more general classical results by E. Noether.

We introduce the general concept of higher order absolute contact differentiation that is based on the idea of semiholonomic contact elements. We clarify how the moving frame method leads to the coordinate functions of the field of r-th order contact elements on a submanifold of Klein space and of the r-th absolute contact differential of a submanifold of Cartan space. We point out that the standard geometric objects of submanifolds are defined on contact elements, so that they are of universal character. In examples, we use heavily the concept of universal horizontal and vertical bundle over contact elements.

This paper is aimed at introducing an algebraic model for physical scales and units of measurement. This goal is achieved by means of the concept of “positive space” and its rational powers. Positive spaces are “semi-vector spaces” on which the group of positive real numbers acts freely and transitively through the scalar multiplication. Their tensor multiplication with vector spaces yields “scaled spaces” that are suitable to describe spaces with physical dimensions mathematically. We also deal with scales regarded as fields over a given background (e.g., spacetime).

Combining an old idea of Olver and Rosenau with the classifica- tion of second and third order homogeneous Hamiltonian operators we classify compatible trios of two-component homogeneous Hamiltonian operators. The trios yield pairs of compatible bi-Hamiltonian opera- tors whose structure is a direct generalization of the bi-Hamiltonian pair of the KdV equation. The bi-Hamiltonian pairs give rise to multi- parametric families of bi-Hamiltonian systems. We recover known ex- amples and we find apparently new integrable systems whose central invariants are non-zero; this shows that new examples are not Miura- trivial.

Software, user guide and examples of computations of integrability related structures for partial differential equations: generalized symmetries, conservation laws, Hamiltonian symplectic and recursion operators.

CDE is a Reduce package devoted to differential-geometric computations on Differential Equations (DEs, for short). The package is included in the official Reduce sources in Sourceforge [37] and it is also distributed on the Geometry of Differential Equations web site http://gdeq.org (GDEQ for short). We start from an installation guide for Linux and Windows. Then we focus on con- crete usage recipes for computations in the geometry of differential equations: higher symmetries, conservation laws, Hamiltonian operators and their Schouten bracket, re- cursion operators. All programs discussed here are shipped together with this manual and can be found in the Reduce sources or at the GDEQ website. The mathematical theory on which computations are based can be found in refs. [12, 22].

A guided tour to REDUCE internals for programmers of REDUCE packages

We prove that the Kupershmidt deformation of a bi-Hamiltonian system is itself bi-Hamiltonian. Moreover, Magri hierarchies of the initial system give rise to Magri hierarchies of Kupershmidt deformations as well. Since Kupershmidt deformations are not written in evolution form, we start with an outline a geometric framework to study Hamiltonian properties of general non-evolution differential equations, developed in Igonin et al. (to appear, 2009) (see also Kersten et al., In: Differential Equations: Geometry, Symmetries and Integrability, Springer, Berlin, 2009).

Abstract. We consider a 3rd-order generalized Monge-Ampere equation tion (which is closely related to the associativity equation in the 2-d topological field theory) and describe all integrable structures related to it (i.e., Hamiltonian, symplectic, and recursion operators). Infinite hierarchies of symmetries and conservation laws are constructed as well.

The phase space of relativistic particle mechanics is defined as the first jet space of motions regarded as time-like one-dimensional submanifolds of spacetime. A Lorentzian metric and an electromagnetic 2-form define naturally a generalized contact structure on the odd-dimensional phase space. In the paper, infinitesimal symmetries of the phase structures are characterized. More precisely, it is proved that all phase infinitesimal symmetries are special Hamiltonian lifts of distinguished conserved quantities on the phase space. It is proved that generators of infinitesimal symmetries constitute a Lie algebra with respect to a special bracket. A momentum map for groups of symmetries of the geometric structures is provided.

We consider the WDVV associativity equations in the four dimensional case. These nonlinear equations of third order can be written as a pair of six component commuting two-dimensional non-diagonalizable hydrodynamic type systems. We prove that these systems possess a compatible pair of local homogeneous Hamiltonian structures of Dubrovin--Novikov type (of first and third order, respectively).

Using the theory of $1+1$ hyperbolic systems we put in perspective the mathematical and geometrical structure of the celebrated circularly polarized waves solutions for isotropic hyperelastic materials determined by Carroll in Acta Mechanica 3 (1967) 167--181. We show that a natural generalization of this class of solutions yields an infinite family of emph{linear} solutions for the equations of isotropic elastodynamics. Moreover, we determine a huge class of hyperbolic partial differential equations having the same property of the shear wave system. Restricting the attention to the usual first order asymptotic approximation of the equations determining transverse waves we provide the complete integration of this system using generalized symmetries.

The theory of Lie remarkable equations, emph{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 $mathbb{R}^k$ and characterize Lie remarkable equations admitted by the considered Lie algebras.

Proceedings of the WASCOM 2011 conference, contributions of the invited speakers. Website of the conference: http://wascom.matematica.unisalento.it/ It is Volume 32 n. 1 of the journal Note di Matematica

Proceedings of the WASCOM 2011 conference. Special issue of the journal Acta Applicandae Mathematicae (ISSN 1572-9036) with the contributions of the speakers. Website of the conference: http://wascom.matematica.unisalento.it/

The Lagrangian representation of multi-Hamiltonian PDEs has been introduced by Y. Nutku and one of us (MVP). In this paper we focus on systems which are (at least) bi-Hamiltonian by a pair $A_1$, $A_2$, where $A_1$ is a hydrodynamic-type Hamiltonian operator. We prove that finding the Lagrangian representation is equivalent to finding a generalized vector field $tau$ such that $A_2=L_tau A_1$. We use this result in order to find the Lagrangian representation when $A_2$ is a homogeneous third-order Hamiltonian operator, although the method that we use can be applied to any other homogeneous Hamiltonian operator. As an example we provide the Lagrangian representation of a WDVV hydrodynamic-type system in $3$ components.

We give a complete description of generalized symmetries and local conservation laws for the fifth-order Karczewska–Rozmej–Rutkowski–Infeld equation for shallow water waves in a channel with variable depth. In particular, we show that this equation has no genuinely generalized symmetries and thus is not symmetry integrable.

We investigate $n$-component systems of conservation laws that possess third-order Hamiltonian structures of differential-geometric type. % Examples include equations of associativity of two-dimensional topological % field theory (WDVV equations). and various equations of Monge-Amp`ere % type. The classification of such systems is reduced to the projective classification of linear congruences of lines in $mathbb{P}^{n+2}$ satisfying additional geometric constraints. Algebraically, the problem can be reformulated as follows: for a vector space $W$ of dimension $n+2$, classify $n$-tuples of skew-symmetric 2-forms $A^{alpha} in Lambda^2(W)$ such that $$ phi_{beta gamma}A^{beta}wedge A^{gamma}=0, $$ for some non-degenerate symmetric $phi$.

The formulation of Geometric Quantization contains several ax- ioms and assumptions. We show that for real polarizations we can generalize the standard geometric quantization procedure by introduc- ing an arbitrary connection on the polarization bundle. The existence of reducible quantum structures leads to considering the class of Liou- ville symplectic manifolds. Our main application of this modified geo- metric quantization scheme is to Quantum Mechanics on Riemannian manifolds. With this method we obtain an energy operator without the scalar curvature term that appears in the standard formulation, thus agreeing with the usual expression found in the Physics literature.

We present a unified mathematical approach for the symbolic com- putation of integrability structures of partial differential equations, like Hamilton- ian operators, recursion operators for symmetries and cosymmetries, symplectic operators. The computations are carried out within the computer algebra system Reduce by the packages CDE and CDIFF.

Let $V$ be a vector space of dimension $n+1$. We demonstrate that $n$-component third-order Hamiltonian operators of differential-geometric type are parametrised by the algebraic variety of elements of rank $n$ in $S^2(Lambda^2V)$ that lie in the kernel of the natural map $S^2(Lambda^2V)to Lambda^4V$. Non-equivalent operators correspond to different orbits of the natural action of $SL(n+1)$. Based on this result, we obtain a classification of such operators for $nleq 4$.