conservation law and generalized Symplectic Monge-Ampere equation arising from 3-variables

If we have a Jacobi PDE system with conservation law $\theta \in \Omega^1(M)$ such that $d \theta$ is non-degenerate 2-form , then we know this fact that it can be written as symplectic 2D Monge-Ampere equation. so if we extend our Jacobi PDE system to 3-forms instead of 2-forms and assume conservation law $\theta \in \Omega^2(M)$ such that 3-form $d \theta$ is non-degenerate 3-form, then the new Jacobi PDE system can be written locally as the generalized 3D Symplectic Monge-Ampere equation arising from 3-forms ?

PS: Firstly, I try to explain more the question and will try to solve it for 2D symplectic monge ampere equation and Jacobi plane and after solve the question for 3D, but I am still looking for an affirmative answer.

Let me explain the Jacobi PDE system view of 2D symplectic Monge Ampere equation .In fact by using Darboux Theorem we can give a nice answer to this question in 2D case . So here, First I prove the following proposition:

But before I try to define Jacobi PDE system for readers as follows

Definition : We call following system as Jacobi PDE system

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

and we can correspond each equation in Jacobi PDE system as symplectic 2-form $\omega$. So we correspond a Jacobi PDE system to two 2-from $\omega_1, \omega_2$ ,also it is clear that the combination of $\omega_1$ and $\omega_2$ are also Jacobi PDE equation, so we can correspond each Jacobi PDE system as $\prod=<\omega_1, \omega_2>$.

Proposition: Let $\prod=<\omega_1,\omega_2>$ be a Jacobi PDE system with a conswerwation law $\theta \in \Omega^1(M)$ , such that $d\theta=a\omega_1+b\omega_2$ is non-degenerate 2-form , then locally the Jacobi PDE system can be written as Monge-Ampere equation.

The reason is: In fact the symplectic Monge-Ampere equation have the following form

$\hat{a}+b\frac{\partial^2\varphi}{\partial x_1^2}-d\frac{\partial^2\varphi}{\partial x_2^2}-c\frac{\partial^2\varphi}{\partial x_2\partial x_1}+e\frac{\partial^2\varphi}{\partial x_1\partial x_2}+\check{a}\frac{\partial\varphi}{\partial x_1}+\tilde{a}\frac{\partial\varphi}{\partial x_2}+$

$f(\frac{\partial^2\varphi}{\partial x_1^2}\frac{\partial^2\varphi}{\partial x_2^2}-\frac{\partial^2\varphi}{\partial x_2\partial x_1}\frac{\partial^2\varphi}{\partial x_1\partial x_2})=0$

where the coefficients are smooth functions of $x$ and $\frac{\partial\varphi}{\partial x}$ .

By the following trick, we reduce the symplectic Monge-Ampere equation into a Jacobi PDE system and vise versa. In fact by substitution $h_1=\frac{\partial\varphi}{\partial x_1}$, and $h_2=\frac{\partial\varphi}{\partial x_2}$ and taking following compatibility condition

$\frac{\partial h_2}{\partial x_1}=\frac{\partial h_1}{\partial x_2}$ we get the following Jacobi PDE system:

$\frac{\partial h_2}{\partial x_1}-\frac{\partial h_1}{\partial x_2}=0$

$a+b\frac{\partial h_1}{\partial x_1}-c\frac{\partial h_1}{\partial x_2}-d\frac{\partial h_2}{\partial x_2}+e\frac{\partial h_2}{\partial x_1}+f(\frac{\partial h_1}{\partial x_1}\frac{\partial h_2}{\partial x_2}-\frac{\partial h_1}{\partial x_2}\frac{\partial h_2}{\partial x_1})=0$

where $a=\hat{a}+\breve{a}h_1+\tilde{a}h_2$ .

Therefore the corresponding 2-forms of this system are:

$\omega_1=dx_1\wedge du_1+dx_2\wedge du_2$

$\omega_2=a(x,u)dx_1\wedge dx_2+ b(x,u)du_1\wedge dx_2+c(x,u)du_1\wedge dx_1+$

$d(x,u)du_2\wedge dx_1 +e(x,u)du_2\wedge dx_2+f(x,u)du_1\wedge du_2$

So the non-degenerate 2-form $d\theta$ , determines a symplectic structure on $M$ .Moreover by applying Darboux theorem, locally there exists a canonical coordinate system for $d\theta$, say $(x_1,x_2,u_1,u_2)$, such that : $d\theta=dx_1\wedge du_1+dx_2\wedge du_2$.

Now, let $\omega'$ be a 2-form, such that $<\omega', d\theta>$ is a local basis for $\prod$ . Then $\omega'$ has the same form as $\omega_2$ as above. Therefore the Jacobi PDE system $d\theta$ and $\omega'$ can be written as the 2D symplectic Monge-Ampere equation as above.

So if we continue this method for 3D symplectic monge ampere equation, we first define 3D Jacobi PDE system(this is my definition for compatibility of 3D Monge-Ampere equation and generalized Jacobi PDE system, please check it yourself again)

Definition (3D Jacobi PDE system): We call following system(4 PDE equations) as a 3D Jacobi PDE system ($k=1,2,3,4$) $a^k(x,h(x))det\pmatrix{\frac{\partial u_1}{\partial x_1} & \frac{\partial u_1}{\partial x_2}&\frac{\partial u_1}{\partial x_3}\cr \frac{\partial u_2}{\partial x_1} & \frac{\partial u_2}{\partial x_2}&\frac{\partial u_2}{\partial x_3}\cr \frac{\partial u_3}{\partial x_1}&\frac{\partial u_3}{\partial x_2}&\frac{\partial u_3}{\partial x_3}\cr} +\sum_{i,j}b^k_{ij}(x,h(x))det \pmatrix{\frac{\partial u_i}{\partial x_1} & \frac{\partial u_i}{\partial x_2}\cr \frac{\partial u_j}{\partial x_1} & \frac{\partial u_j}{\partial x_2}\cr} -$

$\sum_{ij}k^k_{ij}(x,h(x))\pmatrix{\frac{\partial u_i}{\partial x_1} & \frac{\partial u_i}{\partial x_3}\cr \frac{\partial u_j}{\partial x_1} & \frac{\partial u_j}{\partial x_3}\cr}+\sum_{ij}c^k_{ij}(x,h(x))\pmatrix{\frac{\partial u_i}{\partial x_1} & \frac{\partial u_i}{\partial x_3}\cr \frac{\partial u_j}{\partial x_2} & \frac{\partial u_j}{\partial x_3}\cr}+$

$\sum_i m_i^k (x,h(x))\frac{\partial u_i}{\partial x_1}-\sum_i n_i^k (x,h(x))\frac{\partial u_i}{\partial x_2}+\sum_i l_i^k (x,h(x))\frac{\partial u_i}{\partial x_3}+e^k(x,h)=0$

, my computation show that by reducing 3D symplectic monge ampere equation to Jacobi PDE system we have four 3-forms. So if we assume $\prod=<\omega_1,\omega_2,\omega_3,\omega_4>$ be a 3D Jacobi PDE system with a conserwation law $\theta \in \Omega^2(M)$ such that $d\theta=a\omega_1+b\omega_2+c\omega_3+d\omega_4$ be a non-degenerate 3-form. In fact $\omega_1,\omega_2,\omega_3,\omega_4$ have the following 3-forms

$\omega_1=dx_1\wedge dx_2 \wedge du_1+dx_3\wedge dx_2\wedge du_3$

$\omega_2=dx_3\wedge dx_2 \wedge du_2+dx_3\wedge dx_1\wedge du_1$

$\omega_3=dx_1\wedge dx_2 \wedge du_2-dx_3\wedge dx_1\wedge du_3$

$\omega_4=du_1\wedge du_2 \wedge du_3+\sum_{ij}\sum_{k=1}^n a_{ij}^k du_i\wedge du_j\wedge dx_k+$

$\sum_i\sum_{j,k}b_{jk}^i du_i\wedge dx_j \wedge dx_k+dx_1\wedge dx_2 \wedge dx_3$

In fact, $\omega_1,\omega_2, \omega_3$ come of the following compatibility conditions:

$\frac{\partial h_1}{\partial x_2}=\frac{\partial h_2}{\partial x_1}$

$\frac{\partial h_2}{\partial x_3}=\frac{\partial h_3}{\partial x_2}$

$\frac{\partial h_1}{\partial x_3}=\frac{\partial h_3}{\partial x_1}$

So we obtain , the Jacobi PDE system $\prod=<\omega_1,\omega_2,\omega_3,\omega_4>$ can be written as following system:

3-D symplectic monge Ampere equation = $A=a(x,h(x))[\frac{\partial^2\varphi}{\partial x_1^2}\frac{\partial^2\varphi}{\partial x_2^2}\frac{\partial^2\varphi}{\partial x_3^2}+\frac{\partial^2\varphi}{\partial x_1 \partial x_2}\frac{\partial^2\varphi}{\partial x_2 \partial x_3}\frac{\partial^2\varphi}{\partial x_3 \partial x_1}+\frac{\partial^2\varphi}{\partial x_2 \partial x_1}\frac{\partial^2\varphi}{\partial x_3 \partial x_2}\frac{\partial^2\varphi}{\partial x_1 \partial x_3}]-$

$(\frac{\partial^2\varphi}{\partial x_1^2}\frac{\partial^2\varphi}{\partial x_2 \partial x_3}\frac{\partial^2\varphi}{\partial x_3 \partial x_2}+\frac{\partial^2\varphi}{\partial x_2^2}\frac{\partial^2\varphi}{\partial x_1 \partial x_3}\frac{\partial^2\varphi}{\partial x_3 \partial x_1}+\frac{\partial^2\varphi}{\partial x_3^2}\frac{\partial^2\varphi}{\partial x_1 \partial x_2}\frac{\partial^2\varphi}{\partial x_2 \partial x_1})+$

$b(x,h(x))[\frac{\partial^2\varphi}{\partial x_1^2}\frac{\partial^2\varphi}{\partial x_2^2}-\frac{\partial^2\varphi}{\partial x_1 \partial x_2}\frac{\partial^2\varphi}{\partial x_2 \partial x_1}]+c(x,h(x))[\frac{\partial^2\varphi}{\partial x_1^2}\frac{\partial^2\varphi}{\partial x_3^2}-\frac{\partial^2\varphi}{\partial x_1 \partial x_3}\frac{\partial^2\varphi}{\partial x_3 \partial x_1}]+$

$d(x,h(x))[\frac{\partial^2\varphi}{\partial x_2^2}\frac{\partial^2\varphi}{\partial x_3^2}-\frac{\partial^2\varphi}{\partial x_2 \partial x_3}\frac{\partial^2\varphi}{\partial x_3 \partial x_2}]+e(x,h(x))[\frac{\partial^2\varphi}{\partial x_1^2}\frac{\partial^2\varphi}{\partial x_2 \partial x_3}-\frac{\partial^2\varphi}{\partial x_1 \partial x_2}\frac{\partial^2\varphi}{\partial x_1 \partial x_3}]-$

$f(x,h(x))[\frac{\partial^2\varphi}{\partial x_2^2}\frac{\partial^2\varphi}{\partial x_1 \partial x_3}-\frac{\partial^2\varphi}{\partial x_1 \partial x_2}\frac{\partial^2\varphi}{\partial x_2 \partial x_3}]+g(x,h(x))[\frac{\partial^2\varphi}{\partial x_3^2}\frac{\partial^2\varphi}{\partial x_1 \partial x_2}-\frac{\partial^2\varphi}{\partial x_1 \partial x_3}\frac{\partial^2\varphi}{\partial x_2 \partial x_3}]+$

$h'(x,h(x))\frac{\partial^2\varphi}{\partial x_1^2}-i(x,h(x))\frac{\partial^2\varphi}{\partial x_2^2}+j(x,h(x))\frac{\partial^2\varphi}{\partial x_3^2}+k(x,h(x))\frac{\partial^2\varphi}{\partial x_1 \partial x_2}-$

$l(x,h(x))\frac{\partial^2\varphi}{\partial x_2 \partial x_3}+m(x,h(x))\frac{\partial^2\varphi}{\partial x_1 \partial x_3}$

$n(x,h(x))\frac{\partial \varphi}{\partial x_1}+o(x,h(x))\frac{\partial \varphi}{\partial x_2}+p(x,h(x))\frac{\partial \varphi}{\partial x_3}+q(x,h(x))=0$

&

$\frac{\partial^2\varphi}{\partial x_1\partial x_2}=\frac{\partial^2\varphi}{\partial x_2\partial x_1}$

$\frac{\partial^2\varphi}{\partial x_1\partial x_3}=\frac{\partial^2\varphi}{\partial x_3\partial x_1}$

$\frac{\partial^2\varphi}{\partial x_2\partial x_3}=\frac{\partial^2\varphi}{\partial x_3\partial x_2}$

But automatically we have last three equations, because $h_i\in C^1$ so $\varphi\in C^2$ and by substituting these equations in first equation $A$ we get the generic 3D Monge-Ampere equation

-
Your 3D calculations seem to be somewhat off. "Effective" and "primitive" don't make any sense unless you have a symplectic form specified, which I'm guessing is $\Omega=dx_i\wedge du_i$ in this case, so that your $\omega_2=dx_3\wedge\Omega$ and $\omega_3=dx_1\wedge\Omega$ are not $\Omega$-primitive. Moreover, I suspect that your $\omega_1$ has a error in its sign and should be $\omega_1=dx_2\wedge\Omega$. Also, you still have not defined "Jacobi PDE system". Do you just mean the PDE system induced by a set of differential forms, i.e., what we call an exterior differential system? –  Robert Bryant Apr 3 '13 at 20:21
But you still have a sign mistake in your definition of $\omega_1$; the '$-$' in the second term should be '$+$'. Moreover, in your final second order equations, you repeat the first equation twice and don't mention the actual Monge-Ampère equation at all. What you don't seem to realize is that, once you correct this sign, three of your $3$-forms are just $dx_i$-multiples of the $2$-form $\Omega$ that I define in my first comment above and your solutions are actually $\Omega$-Lagrangian, so that the real ideal you are dealing with is actually generated by $\Omega$ and $\omega_4$. –  Robert Bryant Apr 6 '13 at 13:44
Also: You seem to want to define a '3D Jacobi PDE system' as a system of four PDE of the given type (i.e., linear in the minors of the Jacobian matrix with coefficients that depend on the variables $x_i$ and the unknowns $u_i$ (which you are also confusing with the $h_i$, but that's minor)). I think that the reason you want to do this is that you get a system of four equations when you convert a symplectic Monge-Ampère equation (which is second order) to a first order PDE system. However, the system of four first order equations that you get from this conversion is very special. (cont.) –  Robert Bryant Apr 6 '13 at 13:56
(cont.) Most systems of four first order systems of this type are incompatible, so defining a '3D Jacobi PDE system' in your way is going to lead you to study systems, most of which will have no solutions, and this is not good. Instead, I think you should be looking at systems generated by a $2$-form and a $3$-form. More generally, the '$n$D symplectic Monge-Ampère system' should be a system on a manifold of dimension $2n$ that is generated by a $2$-form $\Omega$ and an $\Omega$-primitive $n$-form $\Psi$, while an '$n$D Jacobi PDE system' in dimension $2n$ should be generated by $n$ $n$-forms. –  Robert Bryant Apr 6 '13 at 14:01
@Hassan Jolany: Consider this example that meets your definition of a "3D Jacobi PDE system": Use the notation $h^i_j = \frac{\partial h^i}{\partial x^j}$ and consider the four equations $$h^2_3-h^3_2-x^1=h^3_1-h^1_3-x^2=h^1_2-h^2_1-x^3=h^1_1+h^2_2+h^3_3=0.$$ It is trivial to see that there are no solutions to this system; just differentiate Eq.1 with respect to $x^1$, Eq.2 with respect to $x^2$, and Eq.3 with respect to $x^3$, and add them. This system also has a conservation law since, for example, the last equation alone represents a closed nondegenerate $3$-form in your language. –  Robert Bryant Apr 7 '13 at 13:30

I think that you should be careful to define your terms, but let me guess: A Jacobi PDE system for three unknown functions $h_1,h_2,h_3$ of three independent variables $x_1,x_2,x_3$ is a set of three partial differential equations, each of which can be written as a linear combination of the minors (of any rank, including $0$ and $3$) of the Jacobian matrix $J_x(h) = \frac{\partial h}{\partial x}$ with coefficients that are explicit functions of the $6$ variables $x_1,x_2,x_3,h_1,h_2,h_3$. In other words, on $M=\mathbb{R}^6$ (or some open subset) with coordinates $x_1,x_2,x_3,h_1,h_2,h_3$, you have specified three $3$-forms $\Upsilon_i$ ($i=1,2,3$), and a graph $$\Gamma_u = \bigl\lbrace\bigl(x,u(x)\bigr)\ \mid\ x\in D\subseteq\mathbb{R}^3\bigr\rbrace$$ solves your system with $h=u(x)$ for some function $u:D\to\mathbb{R}^3$ if and only if the pullback of each of the $\Upsilon_i$ to $\Gamma_u$ vanishes. To get a decent theory, you'll also need to impose some kind of nondegeneracy condition, such as the condition that, for a 'generic' pair of tangent vectors $v_1,v_2\in T_pM$, the $1$-forms $\theta_i(v) = \Upsilon_i(v_1,v_2,v)$ are linearly independent at $p$.
Now, it turns out that such a system is never equivalent to a so-called symplectic Monge-Ampère system, which is locally described on a $6$-manifold $M$ by a choice of a symplectic $2$-form $\Omega$ on $M$ and a nonzero $3$-form $\Upsilon$ that is $\Omega$-primitive, i.e., such that $\Omega\wedge\Upsilon=0$. (In this case, the $3$-manifolds you want to study are the $\Omega$-Lagrangian submanifolds $L\subset M$ to which $\Upsilon$ pulls back to be the zero $3$-form.
The reason is that for Jacobi PDE systems as above (that satisfy the nondegeneracy condition), the general solution depends locally on three arbitrary functions of $2$-variables while, for the symplectic Monge-Ampère systems as defined above, the general solution depends locally on two arbitrary functions of $2$-variables.
On the other hand, you could consider a generalization of 'symplectic Monge-Ampère system' in which you have a differential ideal $\mathcal{I}$ on a $6$-manifold that is generated by a nondegenerate $2$-form $\Omega$ and a nonvanishing $\Omega$-primitive $3$-form $\Upsilon$. The condition that they generate a differential ideal is just that there exist a $1$-form $\alpha$ and a function $a$ such that $$d\Omega = \alpha\wedge\Omega + a\ \Upsilon.$$ (If $a\equiv0$, then one finds that $d\alpha=0$, so that, at least locally, one can scale $\Omega$ to make it be closed, and you are back in the symplectic Monge-Ampère case. However, if $a\not=0$, you cannot do this.) In this generalized situation, the general solution will depend on two arbitrary functions of $2$ variables, just as in the symplectic Monge-Ampère case. It would now make sense to think of conservation laws as closed $3$-forms in $\mathcal{I}$, and there are interesting question to ask there.