Application Of Invariant Subspace Method To Nonlinear Partial Differential Equations

In this paper, three problems are considered as follows:1. By invariant subspace method, classification of the generalized fourth-order nonlinear differential equation arising from the liquid filmsis described,where F[u] admits the invariant subspace W_n defined by some n-th linear ordinary differential equation with constant coefficientsThe subspaces W_n are on the polynomial, trigonometric, exponential or mixed form. Exact solutions of these equations are constructed on the subspaces W_n. Then these equations are reduced to systems of ordinary differential equations.2. Classification of two-component nonlinear cross-diffusion systemsis presented by a developing invariant subspace method. The vector differential operator (F₁[u,v],F₂[u,v]) admits invariant subspace W_n1¹Ã—W_n2² defined by the system of ordinary differential equationswhere n₁,n₂ = 2,3,4,5. Exact solutions of these systems are constructed. Then these systems are reduced to finite-dimensional dynamical systems.In most cases, two components of the exact solutions belong to different "scalar" subspaces.3. We utilize the method of invariant set related to invariant subspace method to derive the exact solution of two-dimensional reaction-diffusion equations with source termIt is shown that there exists a class of reaction-diffusion equations which are invariant with respect to the sets E₁={u:u_x=v_xf(t)F(u),u_y=v_yf(t)F(u)} and E₂={u:u_x=a'(x)f(t)F(u),u_y=b'(y)g(t)F(u)},with f≠g.As a result, we obtain exact solutions of the certain nonlinear reaction-diffusion equations. These solutions extend the well-known self-similar solutions of the porous medium equation. The behavior to some solutions and the corresponding interfaces are also described.