Jacobi structures of evolutionary partial differential equations

In this paper we introduce the notion of infinite dimensional Jacobi structure to describe the geometrical structure of a class of nonlocal Hamiltonian systems which appear naturally when applying reciprocal transformations to Hamiltonian evolutionary PDEs. We prove that our class of infinite dimensional Jacobi structures is invariant under the action of reciprocal transformations that only change the spatial variable. The main technical tool is in a suitable generalization of the classical Schouten–Nijenhuis bracket to the space of the so called quasi-local multi-vectors, and a simple realization of this structure in the framework of supermanifolds. These constructions are used to compute the Lichnerowicz–Jacobi cohomologies and to prove a Darboux theorem for Jacobi structures with hydrodynamic leading terms. We also introduce the notion of bi-Jacobi structures, and consider the integrability of a system of evolutionary PDEs that possesses a bi-Jacobi structure.     

Synthesis,characterization, electrochemical,catalytic and antimicrobial activity studies of hydrazone Schiff base ruthenium(II) complexes

Four tridentate O, N, O donor Schiff base ligands were prepared by the reaction of substituted benzhydrazide and appropriate salicylaldehyde. The complexes of these ligands were synthesized by refluxing the ligands with ruthenium(II) starting complexes of the formula [RuHCl(CO)(EPh₃)₂B] in benzene, where E = P or As; B = PPh₃ or AsPh₃ or pyridine. The newly synthesized complexes were characterized by elemental, spectral (FT‐IR, UV and NMR) and electrochemical data. On the basis of the above studies, an octahedral structure has been proposed for all the complexes. The catalytic efficiency of the complexes in aryl–aryl couplings and oxidation of alcohols was examined and their inhibition activity against the growth of the micro‐organisms was also examined. Copyright © 2009 John Wiley & Sons, Ltd.     

Chemical shift assignment and conformational analysis of monoalkylated acylguanidines

Monoalkylated acylguanidines are important functional groups in many biologically active compounds and additionally applied in coordination chemistry. Yet a straightforward assignment of the individual NH chemical shifts and the acylguanidine conformations is still missing. Therefore, in this study, NMR spectroscopic approaches for the chemical and especially the conformational assignment of protonated monoalkylated acylguanidines are presented. While NOESY and ³J_{H, H} scalar couplings cannot be applied successfully for the assignment of acylguanidines, ⁴J_{H, H} scalar couplings in ¹H,¹H COSY spectra allow for an unambiguous chemical shift and conformational assignment. It is shown that these ⁴J_{H, H} long‐range couplings between individual acylguanidinium NH resonances are observed solely across all‐trans (w) pathways. Already one cis orientation in the magnetisation transfer pathway leads to signal intensities below the actual detection limit and significantly lower than cross‐peaks from ²J_{NH, NH} couplings or chemical exchange. However, it should be noted that also in the case of conformational exchange being fast on the NMR time scale, averaged cross‐peaks from all‐trans ⁴J_{H, H} scalar couplings are detected, which may lead at first glance to an incomplete or even wrong conformational analysis. Copyright © 2010 John Wiley & Sons, Ltd.     

Anti-self-dual Yang–Mills equations on noncommutative space–time

By replacing the ordinary product with the so-called -product, one can construct an analog of the anti-self-dual Yang–Mills (ASDYM) equations on the noncommutative . Many properties of the ordinary ASDYM equations turn out to be inherited by the -product ASDYM equation. In particular, the twistorial interpretation of the ordinary ASDYM equations can be extended to the noncommutative , from which one can also derive the fundamental structures for integrability such as a zero-curvature representation, an associated linear system, the Riemann–Hilbert problem, etc. These properties are further preserved under dimensional reduction to the principal chiral field model and Hitchin’s Higgs pair equations. However, some structures relying on finite dimensional linear algebra break down in the -product analogs.     

Une modélisation thermomécanique unidimensionnelle de la propagation d'un front de changement de phase dans un monocristal d'AMFA one-dimensional thermomechanical modeling of phase change front propagation in a SMA monocrystal

We propose a behavioral modeling that takes into account the thermomechanical couplings accompanying the phase transition in single-crystal CuZnAl samples. The goal of this model is to put forward the significant role played by the heat diffusion in the propagation mode of the phase change fronts. Numerical simulations showed the existence of such a phase change front and predicted the calorimetric and kinematic effects accompanying its propagation. In particular, an inversion of the propagation way during a creep test and caused by an increase of the room temperature was correctly simulated by the model. To cite this article: A. Chrysochoos et al., C. R. Mecanique 331 (2003).     

Some Bianchi Type III String Cosmological Models with Bulk Viscosity

We investigate the integrability of cosmic strings in Bianchi III space-time in presence of a bulk viscous fluid by applying a new technique. The behavior of the model is reduced to the solution of a single second order nonlinear differential equation. We show that this equation admits an infinite family of solutions. Some physical consequences from these results are also discussed.     

关于一个可积的广义Hamilton方程族

本文利用ｒ－矩阵生成了一个广义的Ｈａｍｉｌｔｏｎ方程族，并证明了它是广义可积的，然后讨论了它和（４）中Ｌｉｏｖｖｉｌｌｅ可积的新的广义Ｈａｍｉｌｔｏｎ方程族之间的关系。     

Time-space integrable decompositions of nonlinear evolution equations

Separation of the time and space variables of evolution equations is analyzed, without using any structure associated with evolution equations. The resulting theory provides techniques for constructing time-space integrable decompositions of evolution equations, which transform an evolution equation into two compatible Liouville integrable ordinary differential equations in the time and space variables. The techniques are applied to the KdV, MKdV and diffusion equations, thereby yielding several new time-space integrable decompositions of these equations.     

From the Toda lattice to the Volterra lattice and back

We discuss the relationship between the multiple Hamiltonian structures of the generalized Toda lattices and that of the generalized Volterra lattices.