Tau Functions Of The Quantum MKP Hierarchy

The study of solitons and integrable systems has been going on for over a hundred years.This research has great significance in physics,algebra and many other disciplines.For over a hundred years,many things have been studied,including KdV equations,the classical KP hierarchy,the discrete KP hierarchy and so on.The mKP hierarchy and quantum mKP(q-mKP)hierarchy are one of the most popular problems in the field of integrable systems in recent years.Q-mKP hierarchy is important in both quantum physics and quantum algebra,so it is a very important integrable hierarchy.Its Lax operator,Lax equation,flow equations,dressing operator and wave function have been studied,but the tau function has not been discussed.This thesis aims to give the bilinear equation and tau functions of q-mKP hierarchy.Firstly,the history of integrable systems is introduced.Then the current research situation of mKP hierarchy is introduced,including its Lax operator,flow equation,wave function,tau function and Fay equation.Then q-differential is introduced into mKP hierarchy to discuss tau function,bilinear equation and Fay equation of q-mkp hierarchy.The existence of tau function makes Lax equation to be solved.It is the basis of the integrable natures,such as gauge transformation and algebraic constraints.The results can not only provide some ties between integrable systems and quantum algebras,but also enrich the theory of the integrable system.Therefore,the research on tau function is meaningful.