Doubt in Peter Olver “Applications of Lie groups to differential equations”

Question

Book: Applications of Lie groups to differential equations. Second edition (1993).

Page: 117-120. Chapter: 2. Section 2.4: Calculation of symmetry groups.

Example: 2.41. The heat equation.

Question 1: After finding the coefficients $\xi$, $\tau$ and $\phi$ of the general vector field, he finds the six vector fields that constitute the Lie algebra. This is made by indicating that five of the six constants $(c_{1},...)$ are equal to zero and the last one (constant) is made equal to $1$. Why this is made like this? (Or am I misunderstanding?)

Question 2: After this he finds the group to each vector field (equation (2.56)). I don’t understand how $G_{5}$ was found. Especially in the variable $u$: how do I find $u{\cdot}{\exp}(-{\epsilon}x-{\epsilon}^{2}t)$?

P.S.: I found from $G_1$ to $G_4$ by doing: $(\tilde{x},\tilde{t},\tilde{u})={\exp}({{\epsilon}{\textbf{v}}})(x,t,u)$.

Answer 1

1. Those vector fields do not constitute the whole Lie algebra, they just span it (meaning that you get the general element in the Lie algebra by putting the constants back again).

2. The symmetry transformation $G_5$, like the other ones, is found by exponentiation. You compute $(\tilde x(\epsilon),\tilde t(\epsilon),\tilde u(\epsilon))$ from the ODEs $$ \frac{d}{d\epsilon}(\tilde x,\tilde t,\tilde u) = (2 \tilde t,0,-\tilde x \tilde u) $$ with initial condition $(\tilde x(0),\tilde t(0),\tilde u(0))=(x,t,u)$. Clearly $\tilde t(\epsilon) = \text{constant} = t$, and then $\tilde x(\epsilon)=2t\epsilon+x$. (From your question, I understand that may have gotten this far.) Then the final step is solving $$ \frac{d\tilde u}{d\epsilon} = -\tilde x(\epsilon) \tilde u(\epsilon) = -(2t\epsilon+x) \, \tilde u(\epsilon) , $$ which you can do by the method of integrating factor: multiplying by $\exp(t \epsilon^2 + x \epsilon)$ you find $$ \frac{d}{d\epsilon} \left( \tilde u(\epsilon) \, \exp(t \epsilon^2 + x \epsilon) \right) = 0 , $$ so that $$ \tilde u(\epsilon) \exp(t \epsilon^2 + x \epsilon) = \text{constant} = u $$ (using the initial condition $\tilde u(0)=u$ in the last step to identify the constant of integration).