Eur. Phys. J. Plus (2015) 130: 106
https://doi.org/10.1140/epjp/i2015-15106-5

Regular Article

Soliton equations related to the affine Kac-Moody algebra D ₄ ⁽¹⁾

¹ Institute of Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 72 Tzarigradsko chaussee, 1784, Sofia, Bulgaria
² Theoretical Physics Department, Faculty of Physics, Sofia University “St. Kliment Ohridski”, 5 James Bourchier Blvd, 1164, Sofia, Bulgaria
³ Institute of Solid State Physics, Bulgarian Academy of Sciences, 72 Tzarigradsko chaussee, 1784, Sofia, Bulgaria

^* e-mail: gerjikov@inrne.bas.bg

Received: 9 April 2015
Accepted: 5 May 2015
Published online: 8 June 2015

Abstract

We have derived the hierarchy of soliton equations associated with the untwisted affine Kac-Moody algebra D ₄ ⁽¹⁾ by calculating the corresponding recursion operators. The Hamiltonian formulation of the equations from the hierarchy is also considered. As an example we have explicitly presented the first non-trivial member of the hierarchy, which is an one-parameter family of mKdV equations. We have also considered the spectral properties of the Lax operator and introduced a minimal set of scattering data.