Is it possible to solve the below 3-Dimensional Quasli-Linear partial differential equation using the method of characteristics?
$$(y-z)u_x+(z-x)u_y+(x-y)u_z=0 \quad u(x,y,0)=xy$$
So first we want to convert the above PDE into a system of ODE's to solve.
Hence the equivalent system of ODE's is given by:
$$\dot{X}=y-z \quad X(\sigma,0)=\sigma$$ $$\dot{Y}=z-x \quad Y(\sigma,0)=\sigma$$ $$\dot{Z}=x-y \quad Z(\sigma,0)=0$$ $$\dot{U}=0 \quad U(\sigma,0)=X(\sigma,0)Y(\sigma,0)=\sigma^2$$
I am not sure how to determine $X(\sigma,s)$, $Y(\sigma,s)$ and $Z(\sigma,s)$ however. Am I going about this the right way? I would appreciate any help/hints if possible.
As a note, I also tried solving the above PDE using the matrix exponential, but the system of equations generated was too hard to solve.
$$(y-z)u_x+(z-x)u_y+(x-y)u_z=0 \quad u(x,y,0)=xy$$ Charpit-Legendre equations : $$\frac{dx}{y-z}=\frac{dy}{z-x}=\frac{dz}{x-y}=\frac{du}{0}$$ A first characteristic equation comes from $du=0$ : $$u=c_1$$ A second characteristic equation comes from
$\frac{dx}{y-z}=\frac{dy}{z-x}=\frac{dz}{x-y}=\frac{dx+dy+dz}{(y-z)+(z-x)+(x-y)}=\frac{dx+dy+dz}{0}$ $$x+y+z=c_2$$
A third characteristic equation comes from
$\frac{dx}{y-z}=\frac{dy}{z-x}=\frac{dz}{x-y}=\frac{ydx+xdy+zdy+ydz+xdz+zdx}{y(y-z)+x(z-x)+z(z-x)+y(x-y)+x(x-y)+z(y-z)}=\frac{d(xy+yz+zx)}{0}$ $$xy+yz+zx=c_3$$ The general solution of the PDE expressed on the form of implicite equation $c_1=F(c_2,c_3)$ is : $$\boxed{u(x,y,z)=F(x+y+z \:,\: xy+yz+zx)}$$ where $F(X,Y)$ is an arbitrary function of two variables $\begin{cases}X=x+y+z\\Y=xy+yz+zx\end{cases}$
Among the infinite set of solutions they are some sub-sets such as $$u(x,y,z)=f(xy+yz+zx)$$ $$u(x,y,z)=g(x+y+z)$$ and of course many others, for example $u(x,y,z)=h(x^2+y^2+z^2)$ because $(x^2+y^2+z^2)=(x+y+z)^2-2(xy+yz+zx)-1$ .
$f,g,h$ are arbitrary functions.
Boundary condition :
$u(x,y,0)=xy=F(x+y\:,\:xy)=f(xy)$
The function $f(X)$ is determined : $f(X)=X$ that we bring back into the above general solution where $X=xy+yz+zx$. The function : $$\boxed{u(x,y,z)=xy+yz+zx}$$ satisfies the PDE and the boundary condition.