孤子耦合方程族的代数结构、自相溶源和守恒律（英文版）

Themaincontentsofthebookincludethefollowing：Inchapter2，wewouldliketopresentadefinitionofthebi-integrablecouplingsofcontinuousanddiscretesolitonhierarchies，whichcontaintwogivenintegrableequationsastheirsub-systems.Therearemuchrichermathematicalstructuresbehindbi-integrablecouplingsthanscalarintegrableequations.Anditisshownthatsuchbi-integrablecouplingsystemcanpossesszerocurvaturerepresentationandalgebraicstructureassociatedwithsemi-directsumsofLiealgebras.Asapplicationexamplesofthealgebraicstructure，thebi-integrablecouplingsystemoftheMKdVandgeneralizedTodalatticeequationhierarchiesarepresentedfromthistheory.Inchapter3，itisshownthattheKroneckerproductofmatrixLiealgebracanbeappliedtoconstructanewintegrablecouplingsystemandHamiltonianstruc-turesofcontinuousanddiscretesolitonhierarchies.Furthermore，weconstructtheHamiltonianstructureofintegrablecouplingsofsolitonhierarchybyusingtheKro-neckerproduct.ThekeystepsaimatconstructinganewLaxpairsbytheKroneckerproduct.Asillustrateexamples，directapplicationtothecontinuousanddiscretespectralproblemsleadtosomenovelsolitonequationhierarchiesofintegrablecou-plingsystem.Then，wepresenttheHamiltonianstructureofintegrablecouplingsofcontinuousanddiscretehierarchieswiththecomponent-traceidentity.