Symmetry groups and spiral wave solution of a wave propagation equation

We study a third-order nonlinear evolution equation,which can be transformed to the modified KdV equation,using the Lie symmetry method.The Lie point symmetries and the one-dimensional optimal system of the symmetry algebras are determined.Those symmetries are some types of nonlocal symmetries or hidden symmetries of the modified KdV equation.The group-invariant solutions,particularly the travelling wave and spiral wave solutions,are discussed in detail,and a type of spiral wave solution which is smooth in the origin is obtained.     

一类可扩充杆横截挠度方程的Cauchy问题

讨论一类刻划可扩充杆横截挠度的非线性双曲型方程utt+A2u+M(x，‖A1/2 u ‖22)Au=0，这里A=-Δ+I，x∈R\+n，Cauchy问题解的存在唯一性，给出了此 方程有唯一局部解的存在定理.文章所给出的结果的适用性要远大于已有的与此问题相关的结论，对此方程非线性 项的假设要比一般的多.事实上，文章的结果是在打破了以前的所有限制而得到的.      

直接法的推广与广义Burgers方程的条件对称约化

A generalization of the direct method of Clarkson and Kruskal for finding similarity reductions of partial differential equations with arbitrary functions is found and discussed for the generalized Burgers equation. The corresponding reductions and the exact solutions due to the methods of the ordinary differential equations are then given by the methods. The results given here answer partially an open problem proposed by Clarkson, that is how to develop the direct method to seek symmetry reductions of nonlinear PDEs with arbitrary functions.