Abstract

We will present real Hamiltonian forms of 2-dimensional Toda  field theories related to exceptional simple Lie algebras [1], and the spectral theory of the associated Lax operators. Real Hamiltonian forms [2] are a special type of “reductions” of Hamiltonian systems, similar to real forms of semi-simple Lie algebras. Examples of real Hamiltonian forms of affine Toda  field theories related to exceptional complex untwisted affine Kac-Moody algebras will be presented. Along with the associated Lax representations, we will also discuss the relevant Riemann- Hilbert problems and derive the minimal sets of scattering data that determine uniquely the scattering matrices and the potentials of the Lax operators.   This is a joint work [3] with Vladimir Gerdjikov and Alexander Stefanov.     References [1] V. S. Gerdjikov and G. G. Grahovski, On reductions and real Hamiltonian forms of affine Toda  eld theories, J Nonlin. Math. Phys. 12 (2005), Suppl. 3, 153-168 V. S. Gerdjikov and G. G. Grahovski, Real Hamiltonian Forms of Affine Toda Models Related to Exceptional Lie Algebras, SIGMA 2 (2006), paper 022 (11 pages)   [2] V. S. Gerdjikov, A. V. Kyuldjiev, G. Marmo and G. Vilasi, Complexifications and Real Forms of Hamiltonian Structures, European J. Phys. B 29 (2002) 177-182; V. S. Gerdjikov, A. V. Kyuldjiev, G. Marmo and G. Vilasi, Real Hamiltonian forms of Hamiltonian systems, European Phys. J. B. 38 (2004) 635-649