Let us consider the Burgers equation 
$$u_t + (u^2)_x  = 0$$
In 

><cite authors="Liu, Tai-Ping; Pierre, Michel">_Liu, Tai-Ping; Pierre, Michel_, [**Source-solutions and asymptotic behavior in conservation laws**](http://dx.doi.org/10.1016/0022-0396(84)90096-2), J. Differ. Equations 51, 419-441 (1984). [ZBL0545.35057](https://zbmath.org/?q=an:0545.35057).</cite>

the authors studied the long-time behavior and showed that the solution converges towards a "N-wave" profile (eq. (2.1)), which is the solution of Burgers with Dirac Delta as initial data. For this, they used the rescaling $u_\lambda = \lambda u(\lambda t, \lambda^2 x)$ and let $\lambda \to 0$.

What happens if we consider instead the Hamilton-Jacobi equation 
$$v_t + (v_x)^2  = 0$$
and study its asymptotic behavior? My guess is that the corresponding "N-wave" should be a primitive of the one for the Burgers equation. I couldn't find this type of result or of computation anywhere though. Can you point to a reference?