Homogeneous Hamiltonian operators and the theory of coverings

A new method (by Kersten, Krasil'shchik and Verbovetsky), based on the theory of differential coverings, allows to relate a system of PDEs with a differential operator in such a way that the operator maps symmetries/conserved quantities into symmetries/conserved quantities of the system of PDEs. When applied to a quasilinear first-order system of PDEs and a Dubrovin-Novikov homogeneous Hamiltonian operator the method yields conditions on the operator and the system that have interesting differential and projective geometric interpretations

Hamiltonian systems of Jordan block type: delta-functional reductions of the kinetic equation for soliton gas

We demonstrate that linear degeneracy is a necessary condition for quasilinear systems of Jordan block type to possess first-order Hamiltonian structures. Multi-Hamiltonian formulation of linearly degenerate systems governing delta-functional reductions of the kinetic equation for dense soliton gas is established (for KdV, sinh-Gordon, hard-rod, Lieb-Liniger, DNLS, and separable cases)

Classification of bi-Hamiltonian pairs extended by isometries

The aim of this article is to classify pairs of first-order Hamiltonian operators of Dubrovin-Novikov type such that one of them has a non-local part defined by an isometry of its leading coefficient. An example of such bi-Hamiltonian pair was recently found for the constant astigmatism equation. We obtain a classification in the case of 2 dependent variables, and a significant new example that is an extension of a hydrodynamic type system obtained from a particular solution of the WDVV equations

Two Dimensional Finite Difference Model with a Singularity Attenuation Factor for Structural Health Monitoring of Single Lap Joints

A Finite Difference algorithm that evaluates the health conditions of a bonded joint is presented and discussed. The mathematical formulation of the problem is developed paying particular attention to the singularity around the corners of the joint and implementing an original discretisation method of the partial differential equations governing the propagation of the elastic waves. The equations are solved under the only hypothesis of bidimensional field. The algorithm is sensible to defects into the bonded joint and can be used as an effective Structural Health Monitoring tool, as proven by the experiments that show close agreement with the numerical simulations

Projective geometry of homogeneous second order Hamiltonian operators

We prove the invariance of homogeneous second-order Hamiltonian operators under the action of projective reciprocal transformations. We establish a correspondence between such operators in dimension $n$ and $3$-forms in dimension $n + 1$. In this way we classify second order Hamiltonian operators using the known classification of $3$-forms in dimensions $\leq$ 9. Systems of first-order conservation laws that are Hamiltonian with respect to such operators are also explicitly found. The integrability of the systems is discussed in detail