text: The inviscid Burgers' equation is a conservation equation, more generally a first order quasilinear hyperbolic equation. The solution to the equation and along with the initial condition
  can be constructed by the method of characteristics. The characteristic equations are
  Integration of the second equation tells us that  is constant along the characteristic and integration of the first equation shows that the characteristics are straight lines, i.e.,
  where  is the point (or parameter) on the x-axis (t = 0) of the x-t plane from which the characteristic curve is drawn. Since  at -axis is known from the initial condition and the fact that  is unchanged as we move along the characteristic emanating from each point , we write  on each characteristic. Therefore, the family of trajectories of characteristics parametrized by  is
  Thus, the solution is given by
  This is an implicit relation that determines the solution of the inviscid Burgers equation provided characteristics dont intersect. If the characteristics do intersect, then a classical solution to the PDE does not exist and leads to the formation of a shock wave. Whether characteristics can intersect or not depends on the initial condition. In fact, the breaking time before a shock wave can be formed is given by