B.1 Elliptic Functions

The Jacobi elliptic functions are standard forms of elliptic functions [72]. The three basic functions are denoted by sn$(u,k),$cn$(u,k)$ and sn$(u,k)$, where $k$ is the modulus such that $0<k^2<1$. They arise from the inversion of the incomplete elliptic integral of the first kind:

$$u=F(\phi,k)=\int_0^\phi \frac{d\varphi}{\sqrt{1-k^2 \sin^2\varphi}} \quad.$$ (B.1)

The inverse function of $F(u,k)$ is given by the Jacobi amplitude am$(u,k)$

$$\phi= (u,k)=F^{-1}(u,k) \quad.$$ (B.2)

The Jacobi elliptic functions can be defined as follows:

$$(u,k)=\sin\phi=\sin \left(F^{-1}(u,k)\right)=\sin\left(\text{am}(u,k)\right)$$ (B.3)

$$(u,k)=\cos\phi=\cos \left(F^{-1}(u,k)\right)=\cos\left(\text{am}(u,k)\right)$$ (B.4)

$$(u,k)=\sqrt{1-\sin^2\phi}=\sqrt{1-\sin^2\left(\text{am}(u,k)\right)} \quad.$$ (B.5)

By introducing $x=\sin\varphi$ as new integration variable in Eq. (B.1), one obtains the following new expression of the incomplete elliptic integral of the first kind:

$$u=\int_0^s \frac{dx}{\sqrt{(1-k^2x^2)(1-x^2)}} \quad,$$ (B.6)

where $s=\sin\phi$. By taking into account that $u=$sn$^{-1}(\sin\phi,k)$ we may also write the last equation as

$$\int_0^s \frac{dx}{\sqrt{(1-k^2x^2)(1-x^2)}}= ^{-1}(s,k) \quad.$$ (B.7)

Similar formulas can be written for cn$(u,k)$ and dn$(u,k)$. The Jacobi elliptic function are doubly periodic in $K$ and $K'$ in the following sense:

$$(u+2 m K + i2nK',k)=(-1)^{m} (u,k)$$ (B.8)

$$(u+2 m K + i2nK',k)=(-1)^{m+n} (u,k)$$ (B.9)

$$(u+2 m K + i2nK',k)=(-1)^{n} (u,k) \quad,$$ (B.10)

where $m,n\in\mathbb{Z}$, $i=\sqrt{-1}$, $K(k)$ is the complete elliptic integral of the first kind:

$$K(k)=F(\pi/2,k)=\int_0^{\pi/2} \frac{d\varphi}{\sqrt{1-k^2 \sin^2\varphi}}=\int_0^{1} \frac{dx}{\sqrt{(1-k^2 x^2)(1-x^2)}} \quad,$$ (B.11)

$K'(k)=K(k')$ and $k'=\sqrt{1-k^2}$ is the complementary modulus. The Jacobi elliptic functions sn$(u,k),$cn$(u,k),$dn$(u,k)$ can be seen as doubly periodic generalizations of the trigonometric function satisfying the conditions:

$$(u,0)=\sin u$$ (B.12)

$$(u,0)=\cos u$$ (B.13)

$$(u,0)=1 \quad.$$ (B.14)

In addition, they are related through the following identities:

$$^2(u,k)+ ^2(u,k)=1$$ (B.15)

$$k^2 ^2(u,k)+ ^2(i,k)=1$$ (B.16)

$$k^2 ^2(u,k)+k'^2= ^2(u,k)$$ (B.17)

$$^2(u,k)+k'^2 ^2(u,k)= ^2(u,k) \quad.$$ (B.18)