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Let's start with a definition:

Definition: A scalar k-th order differential equation on a smooth manifold $M$, is $F(x,v,\frac{\partial {^\left | \sigma \right |}v}{\partial x^\sigma })=0 $
for $\left | \sigma \right |\leqslant k $.

This equation can be viewed as hypersurface $E=F(x,u,p_\sigma )=0\subset J^kM$ in the space of k-jets.

An n-dimensional integral manifold $L$ of the Cartan distribution is called a multivalued solution of equation E, if $L\subset E$.

(see A.G. Kushner's paper about the classification of Monge-Ampère equations)

How can we find the multivalued solution of the equation $\frac{v_{xx}v_{yy}-v_{xy}^{2}}{(1+v_{x}^{2}+v_{y}^{2})^2}=1 $?

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  • $\begingroup$ What do you mean by "compute"? Are you looking for only solutions that have an explicit formula? $\endgroup$
    – Deane Yang
    Feb 26, 2013 at 21:57
  • $\begingroup$ I am just looking for multivalued solution of this equation, for more information see matematicas.uniandes.edu.co/~mikarm/TALLER_7/pdf/kushner.pdf $\endgroup$
    – user21574
    Feb 26, 2013 at 21:59
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    $\begingroup$ Actually, the final equation asserts that the (multivalued) graph $z = v(x,y)$ has Gauss curvature equal to $1$. Hence, as Lie knew, it is not contact-equivalent to Laplace's equation (though it is, of course, elliptic); it is not even Darboux-integrable (again, Lie). Rather, it is contact equivalent to the elliptic sinh-Gordon equation $\Delta u = \mathrm{sinh}\ u$, which is an integrable equation. Of course, there are many multi-valued solutions, such as $v(x,y)=\pm\sqrt{1-x^2-y^2}$, and many can be found using integrable systems techniques. $\endgroup$
    – Robert Bryant
    Mar 5, 2013 at 13:21
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    $\begingroup$ (correction) What I wrote above about the given equation being contact equivalent to the elliptic sinh-Gordon equation is not correct. Instead, they are Bäcklund equivalent. I forgot that one has to make an integrable extension of the above equation before there is a prolongation equivalence with the elliptic sinh-Gordon equation. The reason is that the above equation does not have any (complex) Riemann invariants until one uses an integrable extension to kill two conservation laws to generate them, and these then become the independent variables in the sinh-Gordon equation. $\endgroup$
    – Robert Bryant
    Mar 5, 2013 at 15:44
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    $\begingroup$ I'm not familiar with Kushner's notes, but whatever he does, it's not going to give you all of the multi-valued solutions explicitly in terms of two arbitrary functions of one variable (or one holomorphic function of one variable), since the equation is not Darboux-integrable. $\endgroup$
    – Robert Bryant
    Mar 11, 2013 at 12:47

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