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

Liu, Tai-Ping; Pierre, Michel, Source-solutions and asymptotic behavior in conservation laws, J. Differ. Equations 51, 419-441 (1984). ZBL0545.35057.

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?