Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
1995-03-29
Physics
High Energy Physics
High Energy Physics - Theory
52 pages, LaTeX. Lectures presented at the VIII J.A. Swieca Summer School, Section: Particles and Fields, Rio de Janeiro - Bra
Scientific paper
Some new developments in constrained Lax integrable systems and their applications to physics are reviewed. After summarizing the tau function construction of the KP hierarchy and the basic concepts of the symmetry of nonlinear equations, more recent ideas dealing with constrained KP models are described. A unifying approach to constrained KP hierarchy based on graded $SL(r+n,n)$ algebra is presented and equivalence formulas are obtained for various pseudo-differential Lax operators appearing in this context. It is then shown how the Toda lattice structure emerges from constrained KP models via canonical Darboux-B\"{a}cklund transformations. These transformations enable us to find simple Wronskian solutions for the underlying tau-functions. We also establish a relation between two-matrix models and constrained Toda lattice systems and derive from this relation expressions for the corresponding partition function.
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