The discrete Toda equation revisited, dual $\beta$-Grothendieck polynomial, ultradiscretization and static soliton

Abstract

This paper presents a study of the discrete Toda equation $({\tau }_n^t{)}^2+{\tau }_{n-1}^t{\tau }_{n+1}^t={\tau }_n^{t-1}{\tau }_n^{t+1}$, that was introduced in 1977. In this paper, it has been proved that the algebraic solution of the discrete Toda equation, obtained via the Lax formalism, is naturally related to the dual Grothendieck polynomial, which is a $K$-theoretic generalization of the Schur polynomial. A tropical permanent solution to the ultradiscrete Toda equation has also been derived. The proposed method gives a tropical algebraic representation of static solitons. Lastly, a new cellular automaton realization of the ultradiscrete Toda equation has been proposed.

15pages

References in corpus (2)

• Duality and deformations of stable Grothendieck polynomials
• Refined dual stable Grothendieck polynomials and generalized Bender-Knuth involutions

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• Closed $k$-Schur Katalan functions as $K$-homology Schubert representatives of the affine Grassmannian
• Jeu de taquin, uniqueness of rectification, and ultradiscrete KP