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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).

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A summary of this set of correspondences is outlined in these lecture notes: 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.

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  • $\begingroup$ Thanks! The heat equation with $t=i\tau$ is the linear Schrodinger with a potential, right? There's no way to get the Burgers equation as a limit/approximation of the nonlinear Schrodinger? $\endgroup$ – Amir Sagiv Oct 2 '17 at 12:08
  • $\begingroup$ Plus, I think there should be a $=\nu \ldots$ instead of $+\nu \ldots$, but I'm not sure $\endgroup$ – Amir Sagiv Oct 2 '17 at 12:13
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    $\begingroup$ yes, thanks, the equals sign was missing, I have corrected it; the Burgers/KPZ equations indeed map onto the linear heat equation (or Schrödinger equation in imaginary time), that's the whole point of this Hopf-Cole transformation: you transform a nonlinear equation into a linear one, which allows for an explicit solution. $\endgroup$ – Carlo Beenakker Oct 2 '17 at 12:26

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