TY - JOUR
T1 - The discrete Toda equation revisited
T2 - Dual β-Grothendieck polynomials, ultradiscretization, and static solitons
AU - Iwao, Shinsuke
AU - Nagai, Hidetomo
N1 - Funding Information:
One of the authors (SI) is partially supported by KAKENHI (26800062). We would like to thank Editage (www.editage.jp) for the English language editing. We are grateful to the anonymous referees for their many helpful suggestions and comments.
Publisher Copyright:
© 2018 IOP Publishing Ltd.
PY - 2018/2/26
Y1 - 2018/2/26
N2 - This paper presents a study of the discrete Toda equation that was introduced in 1977. In this paper, it is proved that the determinantal solution of the discrete Toda equation, obtained via the Lax formalism, is naturally related to the dual Grothendieck polynomials, a K-theoretic generalization of the Schur polynomials. A tropical permanent solution to the ultradiscrete Toda equation is also derived. The proposed method gives a tropical algebraic representation of the static solitons. Lastly, a new cellular automaton realization of the ultradiscrete Toda equation is proposed.
AB - This paper presents a study of the discrete Toda equation that was introduced in 1977. In this paper, it is proved that the determinantal solution of the discrete Toda equation, obtained via the Lax formalism, is naturally related to the dual Grothendieck polynomials, a K-theoretic generalization of the Schur polynomials. A tropical permanent solution to the ultradiscrete Toda equation is also derived. The proposed method gives a tropical algebraic representation of the static solitons. Lastly, a new cellular automaton realization of the ultradiscrete Toda equation is proposed.
KW - cellular automaton
KW - determinant solutions
KW - discrete Toda equation
KW - dual Grothendieck Polynomial
KW - ultradiscrete systems
UR - http://www.scopus.com/inward/record.url?scp=85043506182&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85043506182&partnerID=8YFLogxK
U2 - 10.1088/1751-8121/aaae30
DO - 10.1088/1751-8121/aaae30
M3 - Article
AN - SCOPUS:85043506182
SN - 1751-8113
VL - 51
JO - Journal of Physics A: Mathematical and Theoretical
JF - Journal of Physics A: Mathematical and Theoretical
IS - 13
M1 - 134002
ER -