Solving systems of partial differential equations with functions of different number of variables

I am trying to solve the following system of two partial differential equations

$\partial_t G(x,y,t) + \partial_x G(x,y,t)+\partial_y G(x,y,t) = -i\left[f(x,t) + f(y,t)\right] $

$\partial_t f(x,t) + \partial_x f(x,t) = f(x,t) - iG(x,0,t)$

with initial conditions $f(x,0)=0$ and $G(x,y,0) = G_0(x,y)$ where $G_0$ is a given function (e.g., a double Gaussian packet in the x-y plane)

I tried the following code:

L = 1; T = 1; x0 = -L/4; sigma = L/30; 
sol = NDSolve[{
   D[G[x, y, t], t] == -(D[G[x, y, t], x] + D[G[x, y, t], y]) - I*(f[x, t] + f[y, t]),
    D[f[x, t], t] == -D[f[x, t], x] + f[x, t] - I*(G[x, 0, t]),
    G[x, y, 0] == Exp[-((x - x0)/(Sqrt[2]*sigma))^2 - ((y - x0)/(Sqrt[2]*sigma))^2],

    f[x, 0] == 0
   },
  {G, f}, {x, -L/2, L/2}, {y, -L/2, L/2}, {t, 0, T},
  MaxSteps -> 500]`

But I get the errors:

Function::fpct: Too many parameters in {x,y,t} to be filled from Function[{x,y,t},0][x,t].

Function::fpct: Too many parameters in {x,y,t} to be filled from Function[{x,y,t},0][-0.5,0.].


Function::fpct: Too many parameters in {x,y,t} to be filled from Function[{x,y,t},0][-0.5,0.].

General::stop: Further output of Function::fpct will be suppressed during this calculation.

NDSolve::ndnum: Encountered non-numerical value for a derivative at y == 0.`.

NDSolve::ndnum: Encountered non-numerical value for a derivative at y == 0.`.

Any suggestion on how to proceed?