Considering the very simple electrical dc power flow problem with two nodes, with voltage V1 and V2 respectively, connected via a resistance R, we end up with a system of two nonlinear equations:
$P_1=\frac{V_1(V_1-V_2)}{R}$
$P_2=\frac{V_2(V_2-V_1)}{R}$
We have two equations and four unknowns. I would expect then that, defining two variables (let's say $P_1$ and $P_2$), the system would be solvable for the other two.
However, when using $P_1 = P_2 = 0$, I get infinite solutions where $V_1 = V_2$.
I got curious about this. Is there any property/parameter of nonlinear equation systems that says anything about its solvability/number of solutions?
More specifically, when solving those kind of systems, how do I know the conditions where there will be infinite solutions instead of one or two?
Short answer. There is a unique solution if $P_1 + P_2 \not= 0$. If $P_1 + P_2 = 0$, but $P_1 \not= 0$ (and hence $P_2\not= 0$), then there are no solutions. Finally if $P_1 = P_2 = 0$, there are infinitely many solutions.
Indeed, your system is equivalent to
(1) $RP_1= V_1(V_1-V_2)$ and
(2) $RP_2= -V_2(V_1-V_2)$.
Adding each side of (1) and (2) yields
(3) $R(P_1 + P_2)= (V_1-V_2)^2$
Squaring (1) and (2) and using (3) gives respectively
(4) $R^2P_1^2= V_1^2(V_1-V_2)^2 = V_1^2R(P_1 + P_2)$
and
(5) $R^2P_2^2 = V_2^2(V_1-V_2)^2 = V_2^2R(P_1 + P_2)$
whence, if $P_1 + P_2 \not= 0$, $$ V_1^2 = \frac{RP_1^2}{P_1 + P_2} \quad \text{and}\quad V_2^2 = \frac{RP_2^2}{P_1 + P_2} $$ If now $P_1 + P_2 = 0$, then you get from (3) $V_1 = V_2$. It now follows from (1) and (2) that $P_1 = P_2 = 0$.