Static solutions of a series of coupled pendulums

[Robin04]

Homework Statement

The equation of motions of a series of pendulums coupled by a torsion spring is this:
$\ddot{\Phi_i}=-\frac{k}{ml^2}(2\Phi_i-\Phi_{i-1}-\Phi_{i+1})$, where k is the torsion spring constant, m is the mass of a single pendulum, and l is the length of a single pendulum. We have N pendulums and we define $\Phi_0 = \Phi_1$, and $\Phi_{N+1}=\Phi_N$, in order to avoid the indices to go out of range at the boundaries. I'm looking for static solutions of this system and I'm expecting to get something like a static soliton.

Homework Equations

The Attempt at a Solution

First, for all $i$, $\ddot{\Phi_i} = 0$
Then, to make it a bit simpler in the beginning I tried with N = 3, and all constants are 1.
The equations are:
$\Phi_2-\Phi_1-\sin(\Phi_1)=0$
$\Phi_1+\Phi_3-2\Phi_2-\sin(\Phi_2) = 0$
$\Phi_2-\Phi_3-\sin(\Phi_3)=0$
If i add these equations, I get $\sum_i \sin(\Phi_i)=0$, which is a nicer necessary condition.

I'm struggling to see through this problem. There has to be infinite solutions. I can't give any random value to any $\Phi$ because if I put it into the equations and calculate the rest, my condition for the sums of the sines doesn't work. What tools/tricks can I use to narrow the domain?
I also tried $\Phi_3=\arcsin(-\sin(\Phi_1)-\sin(\Phi_1+\sin(\Phi_1))$ which is a very messy function.

 

[Robin04]

The equation of motions of a series of pendulums coupled by a torsion spring is this:
¨Φi=−kml2(2Φi−Φi−1−Φi+1)

Ops, I made a mistake there. I left out the term from the gravitational potential. The correct equation of motion is:
$\ddot{\Phi_i}=-\frac{k}{ml^2}(2\Phi_i-\Phi_{i-1}-\Phi_{i+1})-\frac{g}{l}\sin(\Phi_i)$

 

[haruspex]

The equations are:
$\Phi_2-\Phi_1-\sin(\Phi_1)=0$
$\Phi_1+\Phi_3-2\Phi_2-\sin(\Phi_2) = 0$
$\Phi_2-\Phi_3-\sin(\Phi_3)=0$

The first eqn implies Φ₁ and Φ₂ have the same sign. Likewise, 2 and 3, so...