Are there any theorems that tell if a particular nonlinear PDE can be solved explicitly by analytical methods?
Where analytical methods I refer to methods such as power series or any methods that use special and elementary functions in some form, as opposed to numerical methods which use iterations or difference schemes etc.
Thanks in advance.
As far as (local) power series solutions go (i.e., in the analytic category) the main existence theorem is the Cauchy-Kowalewski Theorem (in the determined, non characteristic case) and its generalization, the Cartan-Kähler Theorem (in the (possibly overdetermined) involutive case). There are further generalizations that weaken the involutivity hypothesis somewhat, such as $2$-acylcity (in the works of the Spencer school) or weighted involutivity (in the works of the Japanese school), that are useful. In all of these cases, the power series solution can be computed recursively and effectively, so in that sense, the power series solution is 'explicit'.
However, most people don't think of power series solutions as 'explicit analytic solutions'. Instead, what they usually mean is something like writing the classical solution of the wave equation $u_{xy}=0$ in the 'explicit' form $u(x,y) = f(x) + g(y)$, involving 2 arbitrary functions of one variable (each). There are nonlinear equations that admit such representations of solutions, perhaps the most famous being that of the Liouville equation $u_{xy} = e^{2u}$, which can be written in the form $$ u(x,y) = \frac12\log\left(\frac{f'(x)g'(y)}{(f(x)+g(y))^2}\right). $$
However, as Lie showed, the equation $u_{xy} = F(u)$ admits a 'general' solution in the form $$ u(x,y) = U\bigl(x,y,f(x),g(y),f'(x),g'(y),\ldots,f^{(n)}(x),g^{(n)}(y)\bigr) $$ for some given function $U$ of $2n{+}4$ variables with $f$ and $g$ being 'arbitrary' functions of a single variable, only when $F(u) = ae^{bu}$ for some constants $a$ and $b$. Note that, by Lie's theorem, even the linear equation $u_{xy} = u$ does not have such a representation. In fact, equations that admit solutions with such representations are extremely rare, and this is why modern PDE does not attempt to rely on them.
In the 1870s, Gaston Darboux developed a general method for determining when equations of various classes have representations of the above form. For example, Darboux' Method could, in theory (the computations can be formidable), determine when a given second order equation of the form $$ F(x,y,u,u_x,u_y,u_{xx},u_{xy},u_{yy})=0 $$ admits an explicit solution in a form that considerably generalizes Lie's form as described above. This is probably the most advanced result of this kind about explicit analytic representations, even today, and it is quite useful in geometric situations. For example, when Darboux' Method is applied to the minimal surface equation, an elliptic nonlinear equation of the above form, it yields the Weierstrass representation for minimal surfaces in terms of a holomorphic function of a single complex variable.
One place where you can read about Darboux' Method is in a couple of papers by myself, Phillip Griffiths, and Lucas Hsu, Hyperbolic exterior differential systems and their conservation laws, I and II, Selecta Mathematica 1 (1995). For further study of Darboux' Method, there are works by Anderson, Fel, and Vassiliou, among others, that can be profitably consulted. As was suggested in the comments above, if you pursue this, you will eventually need to understand Cartan's Equivalence Method, which is the principal tool needed in the study of geometric invariants of PDE, i.e., properties of PDE that are invariant under all changes of coordinates.
