Invariant Subspace Of First-order With Quadratic Nonlinear Vector Differential Operator And Its Application

In this thesis, invariant subspac.es of first-order quadratic nonlinear vector differ-ential operator are given. Exact solutions of some system of evolution equations with quadratic nonlinearities are constructed on invariant subspaces. Tree problems are considered as follows:(1) Invariant subspaces of first-order quadratic nonlinear vector differential op-erator G[u,v]= (G1[u,v],G2[u,v]) are given, where G1[u,v]=α1uux+α2vvx+α3uvx+α4uxv+α5uv G2[u,v)=β1vvx+β2uux+β3vux+β4vxu+β5uv(2) In the sense of variable transformation, n-dimension Euler equation for poly-tropic gas ρt+▽·(ρu)= 0, (ρu)t+▽·(ρu ⊕ u)+▽P= 0, P=ργ/γ,γ≥1 can be reduced to finite dimensional dynamical systems by the method of invariant subspace method. Then some exact solutions of Euler equation are obtained.(3) By the invariant subspace method, the Euler equations for Chaplygin gas are reduced to finite-dimensional dynamical systems of first-order ordinary differential equations. Some interesting exact solutions, including finite-time blowup solution, global solution and time-periodic solution, are obtained. The unusual blowup inter-faces are given.