On the similarity solution of evolution equation u _t =H(x, t, u, u _x , u _xx , ...)

Abstract

Let

$$\begin{gathered} u^* = u + \in \eta (x,{\text{ }}t,{\text{ }}u), \hfill \\ \hfill \\ \hfill \\ x^* = x + \in \xi (x, t, u{\text{),}} \hfill \\ \hfill \\ \hfill \\ {\text{t}}^{\text{*}} = {\text{ }}t + \in \tau {\text{(}}x,{\text{ }}t,{\text{ }}u), \hfill \\ \end{gathered}$$

be an infinitesimal invariant transformation of the evolution equation u _t =H(x,t,u,∂u/∂x,...,∂ ⁿ:u/∂x ⁿ. In this paper we give an explicit expression for $\eta ^{X^i }$in the ‘determining equation’

$$\eta ^T = \sum\limits_{i = 1}^n {{\text{ }}\eta ^{X^i } {\text{ }}\frac{{\partial H}}{{\partial u_i }} + \eta \frac{{\partial H}}{{\partial u_{} }} + \xi \frac{{\partial H}}{{\partial x}} + \tau } \frac{{\partial H}}{{\partial t}},$$

where u _i =∂ ⁱ u/∂x ⁱ. By using this expression we derive a set of equations with η, ξ, τ as unknown functions and discuss in detail the cases of heat and KdV equations.