Consider some PDE involving a scalar function $u(x,t)$ with two independent variables $x$ and $t$. Assume that this PDE has a Lie Algebra spanned by the following generators,
$X_1=\partial_x,\quad X_2=\partial_t,\quad X_3=\partial_u,\quad \mbox{and} \quad X_4=x\partial_x+2t\partial_t$.
What is the interpretation of $X_1$, $X_2$, and $X_3$? Are these the scaling symmetries? What is the interpreation of $X_4$?
Edit: Nevermind I figured it out. $X_1$, $X_2$, and $X_3$ give the translations of each variable and $X_4$ woulg give $u(2x,3t)$ sort of scaling, right?
For anyone interested the key insight is to exponentiate the generators using the relationship, $\exp(a(\frac{\partial}{\partial x}+\frac{\partial}{\partial t}+\frac{\partial}{\partial u}))u(x,t)=u(x+a,t+a)+a$
Yes, you are correct that $X_{1}, X_{2}, X_{3}$ are transnational symmetries and $X_{4}$ is scaling symmetry. In term of physical relevance we mean PDE is invariant under prolonged $X_{i}$'s(where prolongation order would maximum order of PDE)