New Developments On The Symmetries And Symmetry Reductions Of Nonlinear Equations

Symmetry study plays a very important role in every field of the natural science. Although the symmetry theory is developed quite well in the each branches of Physics and Mathematics, it is still a very vivid and active research field. Especially, in the study of the integrable systems, because of the existence of the infinitely many symmetries, the symmetry studies attract much attentions of the mathematicians and physicists. There are primarily two reasons for this phenomenon. On the one hand, there are still many unsolved important problems in the traditional symmetry theories. On the other hand, one has to develop some new symmetry study methods to investigate the difficult nonlinear systems. In the symmetry study of the (1+1)-dimensional integrable systems, the strong symmetry operator approach is one of the most effective methods. However, there are many hinder in realizing the method in high dimensional systems. In the first part this paper, after reviewing the basic methods of the strong symmetry operators, we successfully established the strong symmetry and the inverse strong symmetry approach to the (3+1)-dimensional Burgers equation. Using this method, infinitely many general symmetries and the full Lie point symmetry algebra of the model are given. Whence a symmetry of a nonlinear system is obtained, the most important problem is how to find some types of exact solutions of the related nonlinear system. Traditionally, to find the exact solutions related to symmetries, one can use either the finite transformation method or the (once) symmetry reduction approach. In the second part of this paper, we firstly reviewed two basic symmetry reduction methods: Clarkson-Kruskal's direct method and the classical Lie group approach. Then we have developed a completely new symmetry reduction method , recurrence reduction approach (RRA). Using the RRA, we can obtain infinitely many new exact solutions and symmetries from a known one symmetry or special solution. To find new types of exact solutions of nonlinear systems is always an important topic in nonlinear science. By using the RRA to the (1+1)-dimensional Burgers equation and the Sharma-Ttass-Olver (STO) equation, various kinds of new exact solutions are obtained. For the (1+1)-dimension Burgers equation, we have obtain one set of generalized rational solutions, one family of rational-kink solutions and two hierarchies of rational-error function solutions. For the STO system, some sets of new types of exact soliton-like and periodic solutions are obtained, such as the rational-negatons, the rational-positons and the rational- complexitons.