2022-05-20T18:30:22Zhttps://repository.dl.itc.u-tokyo.ac.jp/oaioai:repository.dl.itc.u-tokyo.ac.jp:000040292021-03-01T19:47:55Z超離散可積分系の厳密解Exact Solutions of Ultradiscrete Integrable SystemsIWAO, Shinsuke9302410University of Tokyo (東京大学)博士(数理科学)The initial value problem of periodic box-ball systems is solved.Through a limiting procedure called ultradiscretisation, the box-ball system is obtainedfrom a reduced discrete KP equation. Our main theme is ultradiscretising the inverse scatteringmethod for discrete KP.Two key theorems in this thesis are listed below:a) The calculating method for the ultradiscrete limit of Abelian integrals over Riemanniansurfaces is established. A graphical method concerning tropical geometry is the essential toolfor the calculation.b) The explicit expression for the general solution of the reduced KP equation is obtained.The method bases on the inverse sacattering method. It is important to say that our methoddoes not depend on the Fay identity, or on any transcendental equations. In other words,our solutions are constructible.The conbination of these results allows us to analyse the periodic box-ball systems. Ourstrategy to solve the initial value problem is summerised as follows: (i) Lift the initial valueof the BBSs to the initial value of the KP equation. (ii) Solve the initial value problem of theKP equation (Key theorem b). The general solution is given as the combination of Riemanntheta functions. (iii) Ultradiscretise the solution of KP (Key theorem a).thesis2010-03-242010-03-24application/pdfapplication/pdf甲第26106号https://repository.dl.itc.u-tokyo.ac.jp/record/4029/files/IwaoS_22_3_PhD_a.pdfhttps://repository.dl.itc.u-tokyo.ac.jp/record/4029/files/IwaoS_22_3_PhD_b.pdfeng