1. The problem statement, all variables and given/known data Question attached 2. Relevant equations 3. The attempt at a solution So I attempt to just do a), b), c) in one systematic way, however, I'm following examples which I don't quite understand. I have also attached the start of the problem which I'm trying to follow. I started off finding the characteristics which in this case, are dt/dtau = 1 , dx/dtau = -u^2 , du/dtau = 0 solving these, I get t=tau, x=-G(Xi)^2*tau+Xi , u=G(Xi) now here is where I get stuck and confused by the example I'm following, I know how they got the first and last entry, but am confused how they got (1+/- Xi)tau+Xi. If someone can give me a hint/tip on where to go from here, or how to obtain each of the solutions of (-1/2, 1 and 1/2) via the coordinates Xi and tau, would be much appreciated.