Institute for Nuclear Research and Nuclear Energy,
1784 Sofia,
BULGARIA
E-mail: gerjikov@inrne.bas.bg, grah@inrne.bas.bg

Reductions and real Hamiltonian forms of affine Toda field theories

Abstract:
The construction of real Hamiltonian forms (RHF) of Hamiltonian systems, proposed in [1] for dynamical systems, is generalized for 1+1 dimensional integrable models. The basic steps of this construction are: i) complexification of the dynamical variables (and the corresponding Hamiltonian as well); ii) projection on the "real form" with an involutive automorphism which commutes with the complex conjugation and preserves the Hamiltonian.

The systematic construction of RHF is possible for generic (not necessarily integrable) Hamiltonian system but the RHF of any integrable system is again integrable.

Using the reduction techniques of [2], the ${\mathbb Z}_2$-symmetries of the extended Dynkin diagram and the relevant reductions of the ATFT [3] one can classify the RHF of ATFT (for ${\bf D}_r^{(1)}$ ATFT see [4]). The well known ATFT can be viewed as a member of the RHF of these theories.

Each RHF of ATFT allows: 1) Lax representation which determines the symmetry properties of the scattering matrix; 2) hierarchy of Hamiltonian structures generated by a recursion operator. We analyze the relations between the different hierarchies and present explicit nontrivial examples of RHF of  ATFT's.

1. V.S. Gerdjikov, A. Kyuldjiev, G. Marmo, G. Vilasi, Eur. Phys. J. B38 (2004), 635-649.
2. V.S. Gerdjikov, G.G. Grahovski, R.I. Ivanov, N.A. Kostov, Inv. Problems 17 (2001), 999-1015.
3. S.P. Khastgir, R. Sasaki, Progr. Theor. Phys. 95 (1996), 503-518.
4. V.S. Gerdjikov, G.G. Grahovski, J. Nonlin. Math. Phys. (in press).