Symmetry Classifications And Invariant Solutions Of Nonlinear Partial Differential Equation(s)

Symmetry classification of partial diferential equations with arbitrary parameters(or functions) is one of the main applications of classical Lie Theory to diferential equa-tions. The key point to determine the symmetry classification is to find out classificationequations and solve determining equations. The specific forms of these parameters inpractical problems depend on the physics meaning of the system. Lie symmetry classifi-cation is chosen to determine the specific form of arbitrary functions because the majorityof mathematical physics equations have good Lie and non-Lie symmetry groups. Oneof the main problems to be solved in the partial diferential equations is to determine thespecific form of arbitrary function, and obtain the nontrivial symmetry group of the cor-responding equation. But it is very difcult to determine the symmetry classification ofpartial diferential equation(s) except for some trivial cases. In symmetry classification,the symmetry of each parameter should be determined. It is difcult to determine thesymmetry and solve the overdetermined equations of the parameters.The main research work of this article is as follows:Firstly, the Lie symmetry classification of the two-dimensional boundary layer sys-tem with2arbitrary functions(f (u)ux+g(v)ut k1utt+k2px=0, pt=0, ux+vt=0) isgiven, and the invariant solutions of some classifications are obtained.Secondly, the approximate symmetry classification of Kadomtsev-Petviashvili equa-tion ((ut+f (u)σux+uxxx)x+g(x)uyy=0) is given, and the invariant solutions of someclassifications are obtained.At last, the approximate potential symmetry classification of Hopf equation (ut+uux=(K(u)ux)x+εux)is given.