Applications of advances in computer algebra to studying classical integrable systems and related algebraic structures

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The area of my work is the study of differential equations and the development of computer algebra algorithms and programs to investigate these equations. Partial differential equations (PDEs) describe the behaviour of a quantity in space and time. They are especially interesting if they allow solutions that propagate and keep their shape or solutions that have shape preserving structures propagating towards each other, penetrating each other and afterwards taking again their original shape. Such PDEs have a number of unusual properties, for example, they have infinitely many conserved quantities and infinitely many infinitesimal symmetries. They are called 'integrable'. To find these integrable PDEs one formulates mathematical conditions for their special properties in the form of auxiliary equations and tries to solve them. These conditions have in common that they are overdetermined, i.e. they involve more conditions than unknowns. The idea is to have one powerful package of computer programs to solve overdetermined systems and to use that repeatedly to find integrable differential equations of different kind.Work under this grant application has two aims: the further strengthening of the computer algebra package Crack for solving overdetermined systems and its application to integrability problems:- Inversion of Recursion and Hamiltonian Operators- Classification of integrable evolution-type equations with fermionic variables- Integrable ODEs with matrix variables- Bi-hamiltonian structures and Poisson cohomologies of Elliptic AlgebrasSome integrable PDEs do have a practical importance. One can manufacture glass fibers such that light propagation in these fibers is described by the integrable Schroedinger equation. This equation has solutions (propagating pulses of light) that keep their shape, and thus such cables are perfectly suited to transmitting large amounts of data over long distances.