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We know we can reduce Symplectic monge ampere equations to Jacobi PDE system with some compatibility condition. I want to see when the vise versa is correct? and is there any theorem for it.

Here Jacobi PDE system is

$a_1+b_1\frac{\partial h_1}{\partial x_1}-c_1\frac{\partial h_1}{\partial x_2}-d_1\frac{\partial h_2}{\partial x_2}+e_1\frac{\partial h_2}{\partial x_1}+f_1\left (\frac{\partial h_1}{\partial x_1}\frac{\partial h_2}{\partial x_2}-\frac{\partial h_1}{\partial x_2}\frac{\partial h_2}{\partial x_1} \right )=0$ $a_2+b_2\frac{\partial h_1}{\partial x_1}-c_2\frac{\partial h_1}{\partial x_2}-d_2\frac{\partial h_2}{\partial x_2}+e_2\frac{\partial h_2}{\partial x_1}+f_2\left (\frac{\partial h_1}{\partial x_1}\frac{\partial h_2}{\partial x_2}-\frac{\partial h_1}{\partial x_2}\frac{\partial h_2}{\partial x_1} \right )=0$

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Can you give a reference for the definition of 'Jacobi PDE'? This is not standard terminology, as far as I know. If you give a definition, we might be able to help you. –  Robert Bryant Nov 12 '12 at 15:52
    
Dear Robert, I edited it –  Hassan Jolany Nov 12 '12 at 16:26
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1 Answer

up vote 4 down vote accepted

Thanks for the clarification; I wasn't familiar with this terminology. I assume that the coefficients $a_i$, $b_i$, $c_i$, $d_i$, $e_i$, and $f_i$ are specified functions of $x_1,x_2,h_1,h_2$. (Let me know if this is not correct.)

Geometrically, what you have is a pair of $2$-forms on a $4$-dimensional manifold, which you have specified in coordinates $(x_1,x_2,h_1,h_2)$ as $$ \begin{align} \Upsilon_1 &= a_1\ dx_1\wedge dx_2 + b_1\ dh_1\wedge dx_2 + \cdots + e_1\ dh_2\wedge dx_2 + f_1\ dh_1\wedge dh_2\\\\ \Upsilon_2 &= a_2\ dx_1\wedge dx_2 + b_2\ dh_1\wedge dx_2 + \cdots + e_2\ dh_2\wedge dx_2 + f_2\ dh_1\wedge dh_2\ ,\\\\ \end{align} $$ and you are looking for surfaces that are Lagrangian with respect to both $2$-forms, as these correspond to (generalized) solutions to your PDE system.

In order for this system to be 'symplectic', what you want is to find (or, better, determine whether there exist) combinations $\Omega = \lambda^1\ \Upsilon_1 + \lambda^2\ \Upsilon_2$ that satisfy $\Omega^2\not=0$ and $d\Omega =0$. Typically, this is 4 first order linear equations for the two unknown functions $\lambda^i$, so it's overdetermined, and there are 'curvature invariants' that determine when there are solutions and how many there are.

These are what are often called 'conservation laws' in some literature on this subject, and one place you could look (at least in the hyperbolic case; the elliptic case and parabolic cases are somewhat similar, though, at least in the formal setting) is in our paper with Phillip Griffiths and Lucas Hsu, Hyperbolic exterior differential systems and their conservation laws, I & II, which appeared in Selecta Mathematica (New Series) 1 (1995), pp. 21–112 and 265–323. (.dvi files of these papers are available at http://fds.duke.edu/db/aas/math/faculty/bryant/publications.html)

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