Compatible Hamiltonian Operators for the Krichever-Novikov Equation

It has been proved by V. Sokolov that the Krichever-Novikov equation's hierarchy is hamiltonian for the non-local Hamiltonian operator H_0=u_x D^{-1} u_x and possesses twi weakly non-local recursion operatos of degree 4 and 6, L_4 and L_6. We show here that H_0, L_4H_0 and L_6H_0 are compatible Hamiltonian operators for which the Krichever-Novikov equation's hierarchy is hamiltonian

A sufficient condition for a Rational Differential Operator to generate an Integrable System

For a rational differential operator $L=AB^{-1}$, the Lenard-Magri scheme of integrability is a sequence of functions $F_n, n\geq 0$, such that (1) $B(F_{n+1})=A(F_n)$ for all $n \geq 0$ and (2) the functions $B(F_n)$ pairwise commute. We show that, assuming that property $(1)$ holds and that the set of differential orders of $B(F_n)$ is unbounded, property $(2)$ holds if and only if $L$ belongs to a class of rational operators that we call integrable. If we assume moreover that the rational operator $L$ is weakly non-local and preserves a certain splitting of the algebra of functions into even and odd parts, we show that one can always find such a sequence $(F_n)$ starting from any function in Ker B. This result gives some insight in the mechanism of recursion operators, which encode the hierarchies of the corresponding integrable equations

Singular Degree of a Rational Matrix Pseudodifferential Operator

In our previous work, we studied minimal fractional decompositions of a rational matrix pseudodifferential operator: H = AB[superscript -1], where Aand B are matrix differential operators, and B is nondegenerate of minimal possible degree deg(B). In the present paper, we introduce the singular degree sdeg(H)=deg(B), and show that, for an arbitrary rational expression H =∑[subscript α] A[subscript 1][superscript ] (B[subscript 1][superscript α])[superscript -1] ⋯ A[subscript n][superscript α] (B[subscript n][superscript α])[superscript -1], we have sdeg(H) ≤∑[subscript α,i] deg(B[subscript i][superscript α]). If the equality holds, we call such an expression minimal. We study the properties of the singular degree and of minimal rational expressions. These results are important for the computations involved in the Lenard-Magri scheme of integrability

Some remarks on non-commutative principal ideal rings

We prove some algebraic results on the ring of matrix differential operators over a differential field in the generality of non-commutative principal ideal rings. These results are used in the theory of non-local Poisson structures.Comment: 4 page

Validation d'un modèle d'accessibilité par recoupement de données multi-sources. Application aux communes de Belgique

Dans le cadre du projet de recherche MOBLOC (Mobilities and Long Term Location Choice in Belgium), l'exploration à un niveau d'agrégation communal des interactions entre les mobilités résidentielle et quotidienne a nécessité la construction d'un modèle d'accessibilité routière à l'échelle de la Belgique. Dans un premier temps, un modèle de trafic en heures creuses affectant les flux selon un mode tout-ou-rien, est comparé à une base d'observations (MOBEL) ainsi qu'à deux modélisations (Google Maps et un modèle développé à l'UCL par Vandenbulke et al., 2009). Le modèle routier aux heures de pointe du matin procède à l'affectation d'une matrice de demande de déplacement domicile-travail et domicile-études sur la base de la recherche d'un équilibre utilisateur. En complément de l'analyse cartographique, différentes statistiques de qualité d?ajustement sont mises à contribution pour le calibrage et la validation du modèle d'accessibilité MOBLOC.accessibilité routière; heures de pointe; heures creuses; Belgique; modélisation

Rational matrix pseudodifferential operators

The skewfield K(d) of rational pseudodifferential operators over a differential field K is the skewfield of fractions of the algebra of differential operators K[d]. In our previous paper we showed that any H from K(d) has a minimal fractional decomposition H=AB^(-1), where A,B are elements of K[d], B is non-zero, and any common right divisor of A and B is a non-zero element of K. Moreover, any right fractional decomposition of H is obtained by multiplying A and B on the right by the same non-zero element of K[d]. In the present paper we study the ring M_n(K(d)) of nxn matrices over the skewfield K(d). We show that similarly, any H from M_n(K(d)) has a minimal fractional decomposition H=AB^(-1), where A,B are elements of M_n(K[d]), B is non-degenerate, and any common right divisor of A and B is an invertible element of the ring M_n(K[d]). Moreover, any right fractional decomposition of H is obtained by multiplying A and B on the right by the same non-degenerate element of M_n(K [d]). We give several equivalent definitions of the minimal fractional decomposition. These results are applied to the study of maximal isotropicity property, used in the theory of Dirac structures.Comment: 20 page

Some algebraic properties of differential operators

First, we study the subskewfield of rational pseudodifferential operators over a differential field K generated in the skewfield of pseudodifferential operators over K by the subalgebra of all differential operators. Second, we show that the Dieudonne' determinant of a matrix pseudodifferential operator with coefficients in a differential subring A of K lies in the integral closure of A in K, and we give an example of a 2x2 matrix differential operator with coefficients in A whose Dieudonne' determiant does not lie in A.Comment: 15 page

p-reduced multicomponent KP hierarchy and classical W-algebras W(gl_N,p)

For each partition p of an integer N \geq 2, consisting of r parts, an integrable hierarchy of Lax type Hamiltonian PDE has been constructed recently by some of us. In the present paper we show that any tau-function of the p-reduced r-component KP hierarchy produces a solution of this integrable hierarchy. Along the way we provide an algorithm for the explicit construction of the generators of the corresponding classical W-algebra W(gl_N,p), and write down explicit formulas for evolution of these generators along the Hamiltonian flows.Comment: 49 pages, v2: minor editing and corrections following the referee suggestion