The Study Of Two New Integrable Models And A Conjecture

In the first chapter of the thesis, the definition of symmetry is reviewed, then three usual approaches to look for symmetries of a nonlinear equation are simply explained, and at the end of the chapter, three common methods to do symmetry reduction are also briefly illustrated.In the second chapter of the thesis, we do research on the integrability of a new bilinear (2+1)-dimensional Sinh-Gordon (ShG) model equation by means of formal series symmetry approach. In the research, infinite many symmetries and Kac-Moody-Virasoro Lie symmetry algebra of the new model are found, which has actually proved that the new bilinear (2+1)-dimensional ShG model equation is integrable in terms of symmetry. Then, with the general symmetries available above, we also do symmetry reduction of the new model. What interests us very much is that one of the special reduction results is a bilinear (1+1)-dimensional classical Boussinesq equation in fact, which tells us that the first of negative KP hierarchy is not only a (2+1)-dimensional Sinh-Gordon extension, but also a novel (2+1)-dimensional classical Boussinesq extension.Inspired by the two integrable bilinear models in Ref. [41] and the first chapter of the thesis, in the second chapter, we proposed another new (2+1)-dimensional bilinear model, and then proved that the new (2+1)-dimensional bilinear model is also integrable in terms of symmetry with the same formal series symmetry approach. Here, the infinite many symmetries and the corresponding Kac-Moody-Virasoro Lie symmetry algebra of the new model are found, with which, the symmetry reduction of the new model are also done. Finally, combing Prof. Liu Qingping and his collaborators' excellent work on the relevant super-symmetry integrable bilinear models with the three known integrable models in the thesis, a conjecture about a more general integrable bilinear model is proposed.