A geometric realization of the ultradiscrete periodic Toda lattice via tropical plane curves

Abstract: This is a review article on a tropical geometric realization of the ultradiscrete periodic Toda lattice (UD-pTL). Time evolution of the UD-pTL is translated into an addition on the Picard group of its spectral curve, which is a tropical hyperelliptic curve of arbitrary genus depending on the system size. The addition on the Picard group can be realized by using intersection of several tropical plane curves, one of which is the spectral curve. In addition, the tropical eigenvector map, which maps a point in the phase space of the UD-pTL into a set of points on the spectral curve, can also be realized by using intersection of tropical curves. Thus, if the initial values are given then the time evolution of the UD-pTL is completely translated into a motion of intersection points of tropical plane curves. Moreover, all tropical plane curves appearing in the curve intersection are explicitly given in terms of the conserved quantities of the UD-pTL.