Let the PDE be
$$\partial_z \psi = i \partial_x^2 \psi$$
and let $\Gamma = (g(t), h(t))$ be a parametric curve for $t\in [a,b]$, $a<b \in \mathbb{R}$, on which I know $\psi(t)$. For simplicity I also assume that $\Gamma$ has the following properties:
- $g$ and $h$ are differentiable
- $\partial_t g \neq 0 \forall t \in [a,b]$
I want to know if it is possible to solve for $\psi$ given those conditions.
Note: I am aware of Partial Differential Equation with initial condition on a curve and consider that my case is different because it's not the same PDE and the curve is in a more general form. Also that PDE can be solved much more easily than the one I am interested in.
What I tried to do so far is as follows. For one thing I can solve the PDE in Fourier space along the $x$ axis and then transform the solution back to real space which gives (if no errors were made):
$$\psi(x, z) = \sqrt{\frac{1}{4i\pi z}}\int_{-\infty}^{\infty}\psi_0(x-\mu) \left(\exp{\frac{i\mu^2}{4z}}\right) d\mu$$
From this result, I could integrate along $\Gamma$ if I take into consideration that this solution will influence the initial condition. In the image I have this case emphasized. Suppose I take 2 points on the curve, A and B. If I use the value of the initial condition at point A in order to compute the solution I will get a nonzero value when reaching $z=z_B$ at $x = x_B$.
This means that the part of the solution that is computed, in the case described in the image, given a value $z_0$ that corresponds to the interval on which the curve is defined, the solution that arises from the $z<z_0$ region will affect the initial condition on the $z>z_0$ region. So it's not only a basic integration with some offset on the $z$ axis.
My take on solving this issue, since the problem comes from optics rather than mathematics, is that I can assume that all I know about the initial condition is the amplitude along the curve. This way I can add a phase that could compensate the influence of the partial solution as described above. However I did not manage to get to some relevant result with this approach.
Thus my question is if there are some already studied cases that are similar to the problem I have since at this point I am not aware of them. If they are, I would like to know.
EDIT: A friend suggested that I could consider the curve to be a boundary condition, but this did not get me anywhere so far.