The constraint of the Kadomtsev-Petviashvili equation and its special solutions

We construct a local action of the group of rational maps from 2 to GL(n, ℂ), on local solutions of flows of the ZS-AKNS sl(n, ℂ)-hierarchy We show that the actions of simple elements (linear fractional transformations) give local Backlund transformations, and we derive a permutability formula from different factorizations of a quadratic element We prove that the action of simple elements on the vacuum may give either global smooth solutions or solutions with singularities However, the action of the subgroup of the rational maps that satisfy the U(n)-reality condition on the space of global rapidly decaying solutions of the flows in the u(n)-hierarchy is global, and the action of a simple element gives a global Backlund transformation The actions of certain elements in the rational loop group on the vacuum give rise to explicit time-periodic multisolitons (multibreathers) We show that this theory generalizes the classical Backlund theory of the sine-Gordon equation The group structures of Backlund transformations for various hierarchies are determined by their reality conditions We identify the reality conditions (the group structures) for the sl(n, ℝ), u(k,n − k), KdV, Kupershmidt-Wilson, and Gel'fand-Dikii hierarchies The actions of linear fractional transformations that satisfy a reality condition, modulo the center of the group of rational maps, give Backlund and Darboux transformations for the hierarchy defined by the reality condition Since the factorization cannot always be carried out under this reality condition, the action is again local, and Backlund transformations only generate local solutions for these hierarchies unless singular solutions are allowed © 2000 John Wiley & Sons, Inc

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Bäcklund transformations and loop group actions

This paper surveys the classification of integrable evolution equations whose field variables take values in an associative algebra, which includes matrix, Clifford, and group algebra valued systems. A variety of new examples of integrable systems possessing higher order symmetries are presented. Symmetry reductions lead to an associative algebra-valued version of the Painleve transcendent equations. The basic theory of Hamiltonian structures for associative algebra-valued systems is developed and the biHamiltonian structures for several examples are found.

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Integrable Evolution Equations on Associative Algebras

Local conserved densities and zero-curvature conditions for Toda lattice field theories

The authors investigate quantum models on a chain with O(n) and Sp(2k) symmetry. The eigenvalues of the corresponding transfer matrices on a finite lattice are calculated. A generalization of the matrix Bethe ansatz to systems with complicated pseudovacuum is proposed.

Integrable models of quantum one-dimensional magnets with O(n) and Sp(2k) symmetry

We associate a system of integrable, generalised nonlinear Schrodinger (NLS) equations with each Hermitian symmetric space. These NLS equations are considered as reductions of more general systems, this time associated with a reductive homogeneous space. The nonlinear terms are related to the curvature and torsion tensors of the appropriate geometrical space. The Hamiltonian structure is investigated using “r-matrix” techniques and shown to be “canonical” for all these equations. Throughout the reduction procedure this Hamiltonian structure does not degenerate. Each of the above systems of equations is gauge equivalent to a generalised ferromagnet. Reductions of the latter are discussed in terms of the corresponding NLS type equations.

Peter P. Kulish

Papers

Nonlinear Schrödinger equations and simple Lie algebras