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

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’ $$eta ^T = sumlimits_{i = 1}^n {{text{ }}eta ^{X^i } {text{ }}frac{{partial H}}{{partial u_i }} + eta frac{{partial H}}{{partial u_{} }} + xi frac{{partial H}}{{partial x}} + tau } frac{{partial H}}{{partial t}},$$ 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.