Tyurin parameters and elliptic analogue of nonlinear Schr"odinger hierarchy

[摘要] Two``ellipticanalogues''ofthenonlinearSchr"odingerhiererchyareconstructed,andtheirstatusintheGrassmannianperspectiveofsolitonequationsiselucidated.Inadditiontotheusualfields$u,v$,theseellipticanalogueshavenewdynamicalvariablescalled``Tyurinparameters,''whichareconnectedwithafamilyofvectorbundlesovertheellipticcurveinconsideration.Thezero-curvatureequationsofthesesystemsareformulatedbyasequenceof$2imes2$matrices$A_n(z)$,$n=1,2,ldots$,ofellipticfunctions.Inadditiontoafixedpoleat$z=0$,thesematriceshaveseveralextrapoles.Tyurinparametersconsistofthecoordinatesofthosepolesandsomeadditionalparametersthatdescribethestructureof$A_n(z)$'s.Twodistinctsolutionsoftheauxiliarylinearequationsareconstructed,andshowntoformaRiemann-Hilbertpairwithdegenerationpoints.TheRiemann-HilbertpairisusedtodefineamappingtoaninfinitedimensionalGrassmannvariety.TheellipticanaloguesofthenonlinearSchr"odingerhierarchyaretherebymappedtoasimpledynamicalsystemonaspecialsubsetoftheGrassmannvariety.

[学科分类] 数学（综合）

[关键词] soliton equation;elliptic curve;holomorphic bundle;Grassmann variety