The Burger's equation $$y_t (t,x) + y\cdot y_x - y_{xx} =0 \, \, ,$$

can be obtained as a limit of the one-dimensional cubic Nonlinear Schrodinger equation (NLS) $$ i\psi _t (t,x) + \psi _{xx} +|\psi|^2\psi =0 \, \, ,$$

but can also be obtained as an approximation of the Kardar-Parizi-Zhang (KPZ) equation $$h_t (t,x) = h_{xx} + (h_x)^2 +\eta (t,x) \, \, ,$$

where $\eta $ is a Gaussian noise term.

**Question:** I don't know any of the details of both approximations/transformations, nor what kind of results can you deduce from it. Do you know any good references for that?

The Burger's equation Wiki page offers some information, but not enough (for me) to get the full picture. There are a lot of papers which are somehow related to it, but none that I found were good as an introductory text.

Thanks!

**Edit:** I don't want to change the OP, but as noted, the transformation is to a *linear* Schrodinger equation with a potential, not the NLS.

The NLS itself can be obtained as the PDE for the slowly-varying envelope of small perturbations in the KdV and Klein-Gordon equation, see e.g., (Ablowitz, Mark J. Nonlinear dispersive waves: asymptotic analysis and solitons. Vol. 47. Cambridge University Press, 2011).