On The Cohomologies,Deformations And Central Invariants Of Bi-Jacobi Structures

Bi-Jacobi structure is a very important notion that appears in the study of the classification problem of bihamiltonian integrable hierarchies.To study this problem,one needs to characterize the equivalence classes of bihamiltonian integrable hierarchies under Miura type transformations,and this is in turn converted to the study of classification of bihamiltonian structures under Miura type transformations.In the theory of nonlinear integrable systems there is a class of transformations,called reciprocal transformations,which change the spatial and time variables of the evolutionary PDEs and play a crucial role in establishing relations between different integrable hierarchies.So an important problem is to study the classification of bihamiltonian integrable hierarchies under Miura type transformations and reciprocal transformations.However,reciprocal transformations in general destroy the locality of the bihamiltonian structures,so one needs to consider a certain class of nonlocal generalizations of bihamiltonian structures which leads to the notion of bi-Jacobi structures,and this class of bi-Jacobi structures is closed under Miura type transformations and reciprocal transformations.Thus the right form of the above mentioned classification problem is to characterize the equivalence classes of bi-Jacobi integrable hierarchies under Miura type transformations and reciprocal transformations,or equivalently,to classify bi-Jacobi structures under Miura type transformations and reciprocal transformations.This is the problem that we study in this thesis.For a class of semisimple bi-Jacobi structures of hydrodynamic type,we introduce the notion of central invariants for their deformations.We show that this set of central invariants are invariant under Miura type transformations and reciprocal transformations of the deformed bi-Jacobi structures,and propose a conjecture that these central invariants completely characterize the equivalence classes,under Miura type transformations and reciprocal transformations,of deformations of a given semisimple bi-Jacobi structures of hydrodynamic type.The main results of this thesis is a proof of this conjecture for the class of one-component semisimple bi-Jacobi structures of hydrodynamic type.We prove this special case of the conjecture by computing the cohomology groups of the semisimple bi-Jacobi structures of hydrodynamic type,the structures of such cohomology groups enable us to characterize the equivalence classes of deformations of semisimple bi-Jacobi structures of hydrodynamic type under Miura type transformations and reciprocal transformations.Our work develops the methods of computing the cohomology groups for semisimple bihamiltonian structures of hydrodynamic type,and provides an effective tool to study the above conjecture for general semisimple bi-Jacobi structures.