Ultradiscrete hypergeometric solutions to Painlevé cellular automata

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1、ULTRADISCRETEHYPERGEOMETRICSOLUTIONSTOPAINLEVECELLULARAUTOMATA´CHRISORMERODAbstract.WeproposenewsolutionstothePainlev´ecellularautomatathatdonotandcannotbederivedusingtheultradiscretizationmethod.Wecon-sideraliftoftheseequationstoanon-archimedeanvaluationfieldinwhichwemayrelaxthesub

2、tractionfreeframeworkofpreviousexplorationsofthearea.Usingsuchamethod,wederiveasetofhypergeometricsolutionstothePainlev´eequationswhichhaslongbeenthoughtimpossibletoderive.Theareaofintegrablediscretemappingshasblossomedwithinthelastfewyears.Inparticular,theinterestindiscreteversion

3、ofthePainlev´eequations[2].Suchequationsarethoughttobeintegrable,andtherearestandardnotionsofintegra-bilitysuchassingularityconfinement[16]andalgebraicentropy[20].AnalogoustothecontinuousPainlev´eequations,thediscretePainlev´eequationsadmitrationalsolutions[10]andalsohypergeometrics

4、olutions[11].TheultradiscretePainelev´eequationsareobtainedviaalimitingprocesscalledultradiscretization[18].Theultradiscretizationprocesssendsarationalfunctionofsomevariablesf(a1,...,an)toarationalfunctionoverasemiringinanewsetofultradiscretevariablesbythefollowinglimitF(A1,...,An)

5、=limǫlog(f(a1,...,an)ǫ→0wherethenewultradiscretevariablesarerelatedtotheoldvariablesbytheequa-tionsa=eAi/ǫ.Suchaprocesssuccessfullyrelatedanintegrablecellularautomataiknownasthebox-ballsystem[19]tointegrableq-differenceequations[18].Roughlyspeakingitisatransformationbringingthefollo

6、wingbinaryoperationstotheirul-tradiscreteequivalentgivenby(1a)a+b→max(A,B)(1b)ab→A+B(1c)a/b→A−Bwherethereisnoanalogofsubtraction.WederiveanultradiscretePainlev´eequa-tionbyapplyingtheultradiscretizationmethodtospecialsubtractionfreeversionsoftheq-differenceanalogsofthePainlev´eequat

7、ions[6].ThereareultradiscreteanalogsofallsixofthePainlev´eequations[5].Thereisscantevidencethatsuchequationsareintegrable,suchastheexistenceofaLaxPair[15,8]andsomevagueanalogofaversionofsingularityconfinement[9].Thereareknownrationalsolu-tionstosuchequations[17].Thesearetheultradisc

8、retizedsubtractionfreerati