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](https://en.wikipedia.org/wiki/Burgers%27_equation) 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](https://en.wikipedia.org/wiki/Korteweg%E2%80%93de_Vries_equation) and [Klein-Gordon](https://en.wikipedia.org/wiki/Klein%E2%80%93Gordon_equation) equation, see e.g., (Ablowitz, Mark J. Nonlinear dispersive waves: asymptotic analysis and solitons. Vol. 47. Cambridge University Press, 2011).



KPZ-NLS-Burgers relationship
A summary of this set of correspondences is outlined in these [lecture notes:](https://www.uni-muenster.de/imperia/md/content/physik_tp/lectures/ss2017/numerische_Methoden_fuer_komplexe_Systeme_II/burgers.pdf) The three partial differential equations in $x$ and $t$,
$$\text{Burgers:}\;\;\partial u/\partial t+u\partial u/\partial x=\nu\partial^2 u/\partial x^2-\partial U/\partial x$$
$$\text{KPZ:}\;\;\partial h/\partial t-\tfrac{1}{2}(\partial h/\partial x)^2=\nu\partial^2 h/\partial x^2+U$$
$$\text{heat equation:}\;\;\partial\psi/\partial t-\partial^2\psi/\partial x^2=(U/2\nu)\psi$$
are mapped onto each other by the substitutions $u=-\partial h/\partial x$, $\psi=e^{h/2\nu}$. The heat equation is also the Schrödinger equation in imaginary time ($t=i\tau$). The term $U$ plays the role of potential energy. Because it is a linear equation (at least if you take $U$ independent of $\psi$), it can be solved exactly. This is useful, because it allows to test the accuracy of a numerical solution of the nonlinear Burgers and KPZ equations.


