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I am looking for examples of solutions to the following Monge-Ampere equation (also called a determinant Hessian PDE, and note that this is analogous to finding a surface given its Gaussian curvature)

$$\phi_{aa}\phi_{bb}-\phi_{ab}^2 = K(b).$$

An example of a solution to this is $\phi=\sin a\ e^b$ and $K(b)=-e^{2b}$. Are there any other non-trivial (ie non polynomial examples)?

  • Note, one can linearize the above equation using the theory of 2 forms and a partial Legendre transform, but it's not obvious that the solutions may be transformed back in closed form.
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