Combinatorial expressions of the solutions to initial value problems of the discrete and ultradiscrete Toda molecules (Q2852214)

The discrete Toda molecule is a discrete analogue of the Toda molecule [\textit{R. Hirota}, in: Algebraic analysis, Vol. 1, 203--216 (1989; Zbl 0705.35121)] derived using the bilinear formalism; it is a discrete integrable system which possesses a Hankel determinant solution analogous to the Toda molecule. The authors examine the initial value problems of the discrete and ultradiscrete Toda molecules with given initial value. Although it is obvious that the exact value of each dependent variable can be calculated from the initial value in finitely many arithmetic and minimizing operations, it is nontrivial how to formulate the solutions since the equations are nonlinear. Utilizing combinatorial objects, the authors derive an exact expression of the solutions to the initial value problems purely in terms of the initial value.NEWLINENEWLINEFor the discrete Toda molecule, a subtraction-free expression of the solution is derived in terms of non-intersecting paths, for which Flajolet's interpretation of continued fractions and Gessel-Viennot's lemma on determinants are applied. By ultradiscretizing the subtraction-free expression, the solution to the ultradiscrete Toda molecule is obtained. Further combinatorial observations lead to a simpler expression of the solution in terms of shortest paths on a specific graph. Finally, the authors analyze the behavior of the solution obtained in terms of shortest paths in comparison with the box-ball system.