Let $\phi(x,t)$ be smooth function. 


Let $\zeta= x-t$ and $\eta= x+t$. $u=\phi\_\zeta$, $v= \phi\_\eta$.


Let $u$, $v$ satisfies following equations:


1-
$$u\_\eta- v\_\zeta= 0$$
$$v^2u\_\zeta-(1+2uv)u\_\eta+ u^2v\_\eta=0$$


The roles of dependent and independent variables are then interchanged to give


2-$$\zeta\_v-\eta\_u=0$$
$$v^2\eta\_v+(1+2uv)\zeta\_v+u^2\zeta\_u=0$$


Can someone please help me to find the required transformation for getting set 2 from set-1. Is this an example of hodographic transformation. Can someone be precise in explaining this transformation.



hodographic transformation
Another way to understand this 'transformation' is to think in terms of differential forms. Let $(u,v,\eta,\zeta)$ be coordinates on $\mathbb{R}^4$ and consider the pair of $2$-forms
$$
\Upsilon\_1 = du\wedge d\zeta + dv\wedge d\eta
\quad\text{and}\quad
\Upsilon\_2 = v^2 du\wedge d\eta + (1{+}2uv)\ du\wedge d\zeta - u^2 dv\wedge d\zeta.
$$
A surface $S\subset\mathbb{R}^4$ that can be written as a graph in the form $\bigl(u,v,\eta(u,v),\zeta(u,v)\bigr)$ is an integral surface of these two $2$-forms (i.e., they pull back to $S$ to become $0$) if and only if the functions $\bigl(\eta(u,v),\zeta(u,v)\bigr)$ satisfy your second set of equations. If the surface can also be written as a graph in the form $\bigl(u(\eta,\zeta),v(\eta,\zeta),\eta,\zeta\bigr)$, then it is an integral surface of these two $2$-forms if and only if $\bigl(u(\eta,\zeta),v(\eta,\zeta)\bigr)$ satisfy your first set of equations. In that sense, the two equation systems are equivalent. (Note that the forms $\Upsilon\_i$ are linear in $\eta$ and $\zeta$, so one can 'add' solution surfaces to obtain new solution surfaces when they satisfy $du\wedge dv\not=0$.


Thus, each set of equations defines the 'same' set of integral surfaces of the pair of $2$-forms. In some sense, you should regard an integral surface of these $2$-forms as a 'generalized' solution to either set of equations.


Also, looked at in this way, you see that there is no actual `transformation' going on, in the sense that there is no mapping. You are merely changing which variables you want to regard as 'independent', so it's really a change in the way you are describing the same class of surfaces.


**Remark:** While you didn't ask, I should point out that the type analysis of this system shows that it's elliptic in the region where $1+4uv<0$ and hyperbolic in the region where $1+4uv>0$.


