Let $(M,g)$ be a $d$-dimensional Lorentzian manifold, i.e., the metric tensor $g$ has signature $(-1,+1,\dots,+1)$ where $-1$ appears only once and $+1$ appears $(d-1)$ times.
A null hypersurface $\Sigma\subset M$ is one codimension one embedded submanifold whose normal vector $n$ is null, i.e., $$g(n,n)=0.$$
On the other hand a spacelike hypersurface $\Sigma\subset M$ is one codimension one embedded submanifold whose normal vector $n$ is timelike, $$g(n,n)<0.$$
Now, further suppose that $\Sigma$ is globally hyperbolic so that a Cauchy surface can be defined and an initial value problem can be defined. In that sense, there is a theorem stating what is the appropriate initial value on a spacelike hypersurface. This is presented in Wald's General Relativity as:
Theorem 10.1.2: Let $(M,g)$ be a globally hyperbolic spacetime (or a globally hyperbolic region of an arbitrary spacetime) and let $\nabla$ be any derivative operator. Let $\Sigma$ be a smooth, spacelike Cauchy surface. Consider the system of $n$ linear equations for $n$ unknown functions $\phi_1,\dots \phi_n$ of the form $$g^{ab}\nabla_a\nabla_b\phi_i + \sum_j (A_{ij})^a\nabla_a \phi_j + \sum_j B_{ij}\phi_j + C_i = 0\tag{10.1.20}.$$ Then equation (10.1.20) has a well posed initial value formulation on $\Sigma$. More precisely, given arbitrary smooth initial data, $(\phi_i,n^a\nabla_a \phi_i)$ for $i=1,\dots,n$ on $\Sigma$ there exists a unique solution of equation (10.1.20) throughout $M$. Furthermore, the solutions depend continuously on the initial data in the sense described above for the Klein-Gordon equation in flat spacetime. Finally a variation of the initial data outside of a closed subset, $S$, of $\Sigma$ does not affect the solution in $D(S)$ [the domain of dependence].
This tells that given appropriate initial value on a spacelike Cauchy surface, namely, the value of the fields and their first derivatives along the normal, we are granted a unique solution.
This paper alludes to such a result when $\Sigma$ is a null surface, but: (1) it just talks about a single scalar field, (2) it never precisely states a theorem and (3) it is not much rigorous really. The claim is on page 14:
We emphasize that the initial value data needed for the null surface development consists of the function on a pair of null surfaces; it is not the same as the data needed in the standard Cauchy problem, which is the function and its first time derivative on an initial surface [14], [15].
I've tried skimming through the papers [14] and [15] mentioned there, but they never even mention null/lightlike surfaces, so if the result is there it is probably phrased differently and I wasn't able to identify it.
My question: is there one analogue theorem of Theorem (10.1.2) for null surfaces? In other words, a theorem stating what is the appropriate initial value data to be specified on null surfaces to find a unique solution of an equation like (10.1.20)? My particular interest would be in null infinity $\mathscr{I}^\pm$ defined in the Penrose completion of an asymptotically flat spacetime, but I believe such theorem would just care about the surface being null.
References are highly appreciated!
For a single null surface, the existence/uniqueness theorem is not possible. This follows from the consideration in the linear case where if you try to solve the linear wave or Klein-Gordon equation $$ \partial^2_{tt} \phi - \triangle \phi + m^2 \phi = 0 $$ with initial data prescribed along $\{ t = x^1\}$. You easily see that for some initial data no solution exists, while for others infinitely many solutions exist.
In terms of an actual existence and uniqueness theorem when you have more than just a single null surface, quite a bit is known (even in the quasilinear case) rigorously.
I am however pretty sure that the case of data prescribed at general $\mathscr{I}^\pm$ are more delicate, since you are looking at scattering data. (Note that for general AF space-times, the domain of dependence of $\mathscr{I}^\pm$ can in fact be quite small (think any black hole space-time); in those cases existence and uniqueness of solutions from scattering data alone is certainly not true.)