Bi-Presymplectic Representation of Liouville Integrable Systems and Related Separability Theory

Canonical transformations theory for presymplectic systems

Completely Integrable Bi-hamiltonian Systems

We study the geometry of completely integrable bi-Hamiltonian systems, and in particular, the existence of a bi-Hamiltonian structure for a completely integrable Hamiltonian system. We show that under some natural hypothesis, such a structure exists in a neighborhood of an invariant torus if, and only if, the graph of the Hamiltonian function is a hypersurface of translation, relative to the af...

Matrix representation of a sixth order Sturm-Liouville problem and related inverse problem with finite spectrum

‎In this paper‎, ‎we find matrix representation of a class of sixth order Sturm-Liouville problem (SLP) with separated‎, ‎self-adjoint boundary conditions and we show that such SLP have finite spectrum‎. ‎Also for a given matrix eigenvalue problem $HX=lambda VX$‎, ‎where $H$ is a block tridiagonal matrix and $V$ is a block diagonal matrix‎, ‎we find a sixth order boundary value problem of Atkin...

Integrable systems and number theory

where is the derivative of with respect to , was derived to describe water waves in shallow channels [KdV95]. It bears the names of Korteweg and de Vries and has become the prototype integrable nonlinear partial differential equation since the work in [Miu68, MGK68, SG69, Gar71, KMGZ70, GGKM74]. The KdV equation is for the field of integrability what the harmonic oscillator is for quantum mecha...

Liouville Correspondences between Integrable Hierarchies

In this paper, we study explicit correspondences between the integrable Novikov and Sawada–Kotera hierarchies, and between the Degasperis–Procesi and Kaup–Kupershmidt hierarchies. We show how a pair of Liouville transformations between the isospectral problems of the Novikov and Sawada–Kotera equations, and the isospectral problems of the Degasperis–Procesi and Kaup–Kupershmidt equations relate...