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Integrable hierarchies and the modular class

Pantelis A. Damianou, Rui Loja Fernandes (2008)

Annales de l’institut Fourier

It is well-known that the Poisson-Nijenhuis manifolds, introduced by Kosmann-Schwarzbach and Magri form the appropriate setting for studying many classical integrable hierarchies. In order to define the hierarchy, one usually specifies in addition to the Poisson-Nijenhuis manifold a bi-hamiltonian vector field. In this paper we show that to every Poisson-Nijenhuis manifold one can associate a canonical vector field (no extra choices are involved!) which under an appropriate assumption defines an...

Integrable systems and group actions

Eva Miranda (2014)

Open Mathematics

The main purpose of this paper is to present in a unified approach to different results concerning group actions and integrable systems in symplectic, Poisson and contact manifolds. Rigidity problems for integrable systems in these manifolds will be explored from this perspective.

Integrable systems and moduli spaces of rank two vector bundles on a non-hyperelliptic genus 3 curve

Pol Vanhaecke (2005)

Annales de l’institut Fourier

We use the methods that were developed by Adler and van Moerbeke to determine explicit equations for a certain moduli space, that was studied by Narasimhan and Ramanan. Stated briefly it is, for a fixed non-hyperelliptic Riemann surface Γ of genus 3 , the moduli space of semi-stable rank two bundles with trivial determinant on Γ . They showed that it can be realized as a projective variety, more precisely as a quartic hypersurface of ℙ 7 , whose singular locus is the Kummer variety of Γ . We first construct...

Integrable three-dimensional coupled nonlinear dynamical systems related to centrally extended operator Lie algebras and their Lax type three-linearization

J. Golenia, O. Hentosh, A. Prykarpatsky (2007)

Open Mathematics

The Hamiltonian representation for a hierarchy of Lax type equations on a dual space to the Lie algebra of integro-differential operators with matrix coefficients, extended by evolutions for eigenfunctions and adjoint eigenfunctions of the corresponding spectral problems, is obtained via some special Bäcklund transformation. The connection of this hierarchy with integrable by Lax two-dimensional Davey-Stewartson type systems is studied.

Integration of the EPDiff equation by particle methods

Alina Chertock, Philip Du Toit, Jerrold Eldon Marsden (2012)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

The purpose of this paper is to apply particle methods to the numerical solution of the EPDiff equation. The weak solutions of EPDiff are contact discontinuities that carry momentum so that wavefront interactions represent collisions in which momentum is exchanged. This behavior allows for the description of many rich physical applications, but also introduces difficult numerical challenges. We present a particle method for the EPDiff equation that is well-suited for this class of solutions and...

Integration of the EPDiff equation by particle methods∗∗∗∗∗∗

Alina Chertock, Philip Du Toit, Jerrold Eldon Marsden (2012)

ESAIM: Mathematical Modelling and Numerical Analysis

The purpose of this paper is to apply particle methods to the numerical solution of the EPDiff equation. The weak solutions of EPDiff are contact discontinuities that carry momentum so that wavefront interactions represent collisions in which momentum is exchanged. This behavior allows for the description of many rich physical applications, but also introduces difficult numerical challenges. We present a particle method for the EPDiff equation that...

Intensified Doxorubicin-Based Regimen Efficacy in Residual Non-Hodgkin's Lymphoma Disease: Towards a Computationally Supported Treatment Improvement

Y. Kogan, B. Ribba, K. Marron, N. Dahan, V. Vainstein, Z. Agur (2010)

Mathematical Modelling of Natural Phenomena

Despite recent advances, treatment of patients with aggressive Non-Hodgkin's lymphoma (NHL2) has yet to be optimally designed. Notwithstanding the contribution of molecular treatments, intensification of chemotherapeutic regimens may still be beneficial. Hoping to aid in the design of intensified chemotherapy, we put forward a mathematical and computational model that analyses the effect of Doxorubicin on NHL over a wide range of patho-physiological conditions. The model represents tumour growth...

Interaction of Turing and Hopf Modes in the Superdiffusive Brusselator Model Near a Codimension Two Bifurcation Point

J. C. Tzou, A. Bayliss, B.J. Matkowsky, V.A. Volpert (2010)

Mathematical Modelling of Natural Phenomena

Spatiotemporal patterns near a codimension-2 Turing-Hopf point of the one-dimensional superdiffusive Brusselator model are analyzed. The superdiffusive Brusselator model differs from its regular counterpart in that the Laplacian operator of the regular model is replaced by ∂α/∂|ξ|α, 1 < α < 2, an integro-differential operator that reflects the nonlocal behavior of superdiffusion. The order of the operator, α, is a measure of the rate of ...

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