From what I can work out, it seems the global solution would be something like
\begin{equation}
y= z = c_1\cosh(\sqrt{2}mi x) \ \ \text{or} \ \ y= z = c_2\sinh(\sqrt{2}mi x)
\end{equation}
Or some combination of such. I don't see any way a real solution is defined across a continuous region ...
To expand on that, I understand that around x=0, the solutions would be real, or for a constant value of y and z. That's fine in my case, since then my complex field is trivial within my theory and I am happy.
If you mean something else, that real solutions exist that aren't constant, then...
Hi, yeah there is no coupling, hence I also separated the two. In which case, the only solutions would be of the form
\begin{equation}
y = z = K e^{\pm i m x}
\end{equation}
And therefore no real solutions?
As to where it's from, I am trying to prove an equation involving a complex scalar...
The ODE is:
\begin{equation}
(y'(x)^2 - z'(x)^2) + 2m^2( y(x)^2 - z(x)^2) = 0
\end{equation}
Where y(x) and z(x) are real unknown functions of x, m is a constant.
I believe there are complex solutions, as well as the trivial case z(x) = y(x) = 0 , but I cannot find any real solutions. Are...