I am dealing with the solution of the Sine-Gordon differential equation $-\phi_{xx}+\sin\phi=0$ which can be expressed in terms of the Jacobi elliptic sine function $\phi=\arccos\left(2k^{2}\text{sn}^{2}\left(x-\frac{a}{2},k\right)-1\right)$. The modulus $k$ is determined by plugging the above solution into the boundary condition
$\phi_{x}\left(x-an=+0\right)-\phi_{x}\left(x-an=-0\right)=\varepsilon\sin\phi\left(x-an=0\right)$ where $a$ is some constant and $n$ is an integer. I have tried taking the limit of the LHS of the boundary condition by taking a series expansion of the derivative of the solution around $x=an$ to see if I get different results in the $+0$ or $-0$ limit. I am probably doing something wrong as I get LHS to be $0$. I also plugged the solution and boundary condition into Mathematica (code below) and I get $0$ for the LHS of the boundary condition.
Simplify[Limit[D[ArcCos[2 k^2 (JacobiSN[x - a/2, k])^2] - 1, x],
x -> a n, Direction -> 1, Assumptions -> a > 0],
Element[n, Integers]] -
Simplify[Limit[D[ArcCos[2 k^2 (JacobiSN[x - a/2, k])^2] - 1, x],
x -> a n , Direction -> 1, Assumptions -> a > 0],
Element[n, Integers]]
How does one take the limits on the LHS of this boundary condition which has the derivatives of the arccosine of a function of Jacobi elliptic sine function?