Discrete integrable systems and their applications to convergence acceleration algorithms


The (re)discovery of the soliton by N. J. Zabusky and M. D. Kruskal in 1965 in the context of the Fermi-Pasta-Ulam problem initiated a new area of applied mathematics and mathematical physics now known as soliton theory. The soliton is an essentially localized object that may be found in such diverse areas of physics as gravitation and field theory, plasma and solid state physics, and hydrodynamics. Due to the unique interaction properties of solitons, soliton theory has also found important technological applications in optical fibre communication systems. Remarkably, based on the classical geometric work by Backlund and Bianchi [see, for instance, C. Rogers and W. K. Schief, Backlund and Darboux transformations, Cambridge Univ. Press, Cambridge, 2002], these interaction properties had already been observed in 1953 by A. Seeger, H. Donth and A. Kochendorfer in the context of Frenkel and Kontorova's dislocation theory. In mathematical terms, the soliton constitutes a particular solution of special classes of partial differential equations. However, in the past few decades, it has become evident that these ˇ§integrable systemsˇ¨ appear in a variety of guises such as ordinary and partial differential equations, difference and differential-difference equations, cellular automata, and convergence acceleration algorithms. On the other hand, the connection between convergence acceleration algorithms and discrete integrable systems is a subject whose interest is rapidly growing among workers in the field.

The focus of this research includes two parts: the first part is on the integrable bilinear study of the Heisenberg-type equations and their discrete analogues. The emphasis is placed on integrable discretizations of some Heisenberg-type equations such as the so-called Myrzakulov equations. The second part is to investigate some properties of discrete integrable systems that are related to the properties of convergence acceleration algorithms. We expect that discretizing integrable PDEs will lead to new sequence transformations which may further be studied from the viewpoint of their algebraic and acceleration properties. Also the connections between some discrete integrable systems and convergence acceleration algorithms will be investigated in more detail.
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