I am reading a set of lecture notes describing the process of symmetry reduction of differential equations via Lie algebra methods (link), and I'm stuck at what seems to be a fairly simple point.
The specific example is the Korteweg-DeVries equation,
$$u_t+uu_x+u_{xxx}=0.$$
The Lie algebra will be realized as a vector field
$$\hat{v}=\xi(x,t,u) \partial_x+\tau(x,t,u) \partial_t+\phi(x,t,u)\partial_u.$$
Now I'll directly cut from the notes:
(I am excerpting but I don't think I'm missing anything critical - check out pg 5 of that link if you want more background).
Anyway, we go through the process of determining the vector field, and end up with
$$\xi=1+t+x,\qquad \tau=1+3t,\qquad \phi=-2u+1$$
So now, I should be integrating these equations, subject to the initial conditions $\tilde{x}(0)=x$, etc. That shouldn't be a problem:
$$\frac{d\tilde{x}}{d\lambda}=1+\tilde{t}+\tilde{x}\rightarrow \tilde{x}=e^\lambda(1+\tilde{t}+x)-(1+\tilde{t})$$ $$\frac{d\tilde{t}}{d\lambda}=1+3\tilde {t}\rightarrow \tilde{t}=\frac{1}{3}e^{3\lambda}(1+3t)-\frac{1}{3}$$ $$\frac{d\tilde{u}}{d\lambda}=-2\tilde{u}+1\rightarrow \tilde{u}=\frac{1}{2}e^{-2\lambda}(2u+1)-\frac{1}{2}$$ (we could eliminate $\tilde{t}$ from the first equation, but that won't help what I'm going to say next)
The problem is that in the notes (pg 9), the result is given as
So for this to match my results I would need to take, for instance,
$$t_0=\frac{1}{3}(1-e^{-3\lambda}),$$
certainly not a constant.
So what am I doing wrong? I guess I am concerned with not understanding what "composing the one-dimensional subgroups" means, but I thought that is describing the process of getting from one solution to another. Like once you get $u(x,t)$, you can use the results above to get another solution $\tilde{u}(\tilde{x},\tilde{t})$.