Integrable 6-Bose System/ Integrable Lattice Hierarquies

Abstract

1)The 2M-Bose integrable model has been continually studied by the IFT-Unesp group in collaboration with Prof. Aratyn from the University of Illinois in Chicago for several values of M, and at on of its simplest slopes it reveals itself obeying the conjecture proposed by Ablowitz, Ramani and Segur, where an integrable system reduces at a self-similarity limit to a system of equations with the Painleve property.In the already studied cases, it was observed a reduction not only to the traditional Painleve equations, but also to new mixed equations, whose solutions and behavior are interest targets.In the case of this project, the aim is the 6-Bose model which has already shown to be reducible to a PVI equation, however there are possible reductions to mixed equations of higher order, as was already observed at the 4-bose model, and these equations must be object of careful analysis over its symmetry properties and origin with respect to its corresponding symmetry groups.The study of rational solutions for the PVI equation by the Pade approximant method will be addressed by our group, where there will be a collaboration from the master's degree student Gabriel Lobo (IFT-Unesp) as well. A comparison with the relevant work of Mozzocco will be useful.2)First will be studied the lattice Sine-Gordon model and its quantization as an introductory model, as a goal of to merge into the current situation of research about these problems and then go to a possible use of such knowledge for other aspects of the problem.The Ablowitz-Ladik hierarchy will be studied too.