MHB Is This Equation Known as the Korteweg-de Vries Equation?

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\begin{align*}
\psi_t + \psi_{xxx} + f(\psi)\psi_x &= 0
\end{align*}

This equation leads to the nonlinear Shrodinger equation but does this equation have a name?
 
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Are there any good visualization tutorials, written or video, that show graphically how separation of variables works? I particularly have the time-independent Schrodinger Equation in mind. There are hundreds of demonstrations out there which essentially distill to copies of one another. However I am trying to visualize in my mind how this process looks graphically - for example plotting t on one axis and x on the other for f(x,t). I have seen other good visual representations of...
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