Symplectic aspects of the tt∗-Toda equations

Ryosuke Odoi

doi:10.1088/1751-8121/ac59ff

Symplectic aspects of the tt^∗-Toda equations

Ryosuke Odoi^*

^*Corresponding author for this work

School of Fundamental Science and Engineering

Research output: Contribution to journal › Article › peer-review

Abstract

We evaluate explicitly, in terms of the asymptotic data, the ratio of the constant pre-factors in the large and small x asymptotics of the tau functions for global solutions of the tt∗-Toda equations. This constant problem for the sinh-Gordon equation, which is the case n = 1 of the tt∗-Toda equations, was solved by Tracy (1991 Commun. Math. Phys. 142 297-311). We also introduce natural symplectic structures on the space of asymptotic data and on the space of monodromy data for a wider class of solutions, and show that these symplectic structures are preserved by the Riemann-Hilbert correspondence.

Original language	English
Article number	165201
Journal	Journal of Physics A: Mathematical and Theoretical
Volume	55
Issue number	16
DOIs	https://doi.org/10.1088/1751-8121/ac59ff
Publication status	Published - 2022 Apr 22

Keywords

Riemann-Hilbert correspondence
ttequations
τ-function

ASJC Scopus subject areas

Statistical and Nonlinear Physics
Statistics and Probability
Modelling and Simulation
Mathematical Physics
Physics and Astronomy(all)

Access to Document

10.1088/1751-8121/ac59ff

Cite this

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title = "Symplectic aspects of the tt∗-Toda equations",

abstract = "We evaluate explicitly, in terms of the asymptotic data, the ratio of the constant pre-factors in the large and small x asymptotics of the tau functions for global solutions of the tt∗-Toda equations. This constant problem for the sinh-Gordon equation, which is the case n = 1 of the tt∗-Toda equations, was solved by Tracy (1991 Commun. Math. Phys. 142 297-311). We also introduce natural symplectic structures on the space of asymptotic data and on the space of monodromy data for a wider class of solutions, and show that these symplectic structures are preserved by the Riemann-Hilbert correspondence. ",

keywords = "Riemann-Hilbert correspondence, ttequations, τ-function",

author = "Ryosuke Odoi",

note = "Funding Information: This work is part of the author{\textquoteright}s PhD thesis at Waseda University. He would like to acknowledge his supervisor, Professor Martin Guest for his support throughout this work and his guidance in this field. He would like to acknowledge Professor Alexander Its for his friendly advice and his suggestions regarding the constant problem. He is also glad to acknowledge the financial support from the mathematics and physics unit {\textquoteleft}multiscale analysis, modelling and simulation{\textquoteright}, top global university project, Waseda University. Publisher Copyright: {\textcopyright} 2022 IOP Publishing Ltd.",

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N1 - Funding Information: This work is part of the author’s PhD thesis at Waseda University. He would like to acknowledge his supervisor, Professor Martin Guest for his support throughout this work and his guidance in this field. He would like to acknowledge Professor Alexander Its for his friendly advice and his suggestions regarding the constant problem. He is also glad to acknowledge the financial support from the mathematics and physics unit ‘multiscale analysis, modelling and simulation’, top global university project, Waseda University. Publisher Copyright: © 2022 IOP Publishing Ltd.

PY - 2022/4/22

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N2 - We evaluate explicitly, in terms of the asymptotic data, the ratio of the constant pre-factors in the large and small x asymptotics of the tau functions for global solutions of the tt∗-Toda equations. This constant problem for the sinh-Gordon equation, which is the case n = 1 of the tt∗-Toda equations, was solved by Tracy (1991 Commun. Math. Phys. 142 297-311). We also introduce natural symplectic structures on the space of asymptotic data and on the space of monodromy data for a wider class of solutions, and show that these symplectic structures are preserved by the Riemann-Hilbert correspondence.

AB - We evaluate explicitly, in terms of the asymptotic data, the ratio of the constant pre-factors in the large and small x asymptotics of the tau functions for global solutions of the tt∗-Toda equations. This constant problem for the sinh-Gordon equation, which is the case n = 1 of the tt∗-Toda equations, was solved by Tracy (1991 Commun. Math. Phys. 142 297-311). We also introduce natural symplectic structures on the space of asymptotic data and on the space of monodromy data for a wider class of solutions, and show that these symplectic structures are preserved by the Riemann-Hilbert correspondence.

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