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.
- Y. He and H.W. Tam, ˇ§Bilinear Backlund transformation and Lax pair for a coupled Ramani equation,ˇ¨ Journal of Math. Anal. and Applications, Vol. 357, Issue 1, pp. 132-136, September 2009.
- Y.F. Zhang and H.W. Tam, ˇ§Coupling commutator pairs and integrable systems,ˇ¨ Chaos, Solitons, and Fractals, Vol. 39, Issue 3, pp. 1109-1120, Feb 2009.
- Y.P. Sun and H.W. Tam, ˇ§Grammian solutions and Pfaffianization of a non-isospectral and variable-coefficient Kadomtsev-Petviashvili equation,ˇ¨ Journal of Math. Anal. and Applications, Vol. 343, Issue 2, pp. 810-817, July 2008.
- H.W. Tam and Z.N. Zhu, ˇ§(2+1)- dimensional integrable lattice hierarchies related to discrete fourth-order non-isospectral problems,ˇ¨ Journal of Physics A: Math Theo, Vol. 40, No. 43, pp. 13031-13045, Oct 2007.
- Y.P. Sun and H.W. Tam, ˇ§A hierarchy of non-isospectral multi-component AKNS equations and its integrable couplings,ˇ¨ Physics Letters A, Vol. 370, Issue 2, pp. 139-144, Oct 2007.