报告摘要： There are many wellknown techniques for obtaining exact solutions of differential equations, but most of them are merely special cases of a few powerful symmetry methods. These methods can be applied to nonlinear differential equations of an unfamiliar type and do not rely on particular tricks. In this talk, we give a straightforward introduction and applications of symmetry methods. For a higherorder ODE, a correspondence between first integrals and invariance under point symmetries holds only when the ODE has a variational principle. If a system of PDEs is invariant under a Lie group of point transformations, one can find, constructively, special solutions, called similarity solutions or invariant solutions that are invariant under a subgroup of the full group admitted by the system. We will illustrate these techniques to investigate a degenerate parabolic system which arises in biological population, and obtain bounded wave solutions under certain conditions.
