Transformações de Bäcklund para hierarquias integráveis abelianas - Retore, A.L [UNESP] (2015)
We study the construction of integrable hierarchies. These hierarchies have inﬁnite equations of motion which arise from the same algebraic structure, and, as a consequence, we can ﬁnd simultaneously and systematically its solitonic solutions using the Dressing method. Inthiswork, we study the mKdV and KdV hierarchies and calculate explicitly the ﬁrst few equations of motion for both of them. To the KdV, the Lax operator seems to work only in positive degrees. We determine the Bäcklund Transformations to the positive degrees of mKdV and KdV hierarchies using the fact that equations of motion can be written as total derivatives. We obtain a systematic way to construct the Bäcklund Transformations for the equations of the mKdV hierarchy exploring the gauge invariance of zero curvature equation. We determine the Bäcklund Transformations of Type-I and Type-II for the odd-degrees equations of mKdV hierarchy. We make the explicit calculation for ﬁrst three positive degrees and also for the next three negative ones