Table 1 The exact solutions of the Jacobi elliptic differential equation ( 12 ) when ${\mathit{e}}_{\mathbf{0}}$ , ${\mathit{e}}_{\mathbf{1}}$ and ${\mathit{e}}_{\mathbf{2}}$ take special values

1

$-(1+m^2)$

$m^2$

sn(ξ) or cd(ξ)

$1-m^2$

$2m^2-1$

$-m^2$

cn(ξ)

$m^2-1$

$2-m^2$

−1

dn(ξ)

$m^2$

$-(m^2+1)$

1

ns(ξ) or dc(ξ)

$-m^2$

$2m^2-1$

$1-m^2$

nc(ξ)

−1

$2-m^2$

$m^2-1$

nd(ξ)

$1-m^2$

$2-m^2$

1

cs(ξ)

1

$2-m^2$

$1-m^2$

sc(ξ)

1

$2m^2-1$

$m^2(m^2-1)$

sd(ξ)

$m^2(m^2-1)$

$2m^2-1$

1

ds(ξ)

$\frac{1}{4}$

$\frac{1}{2}(1-2m^2)$

$\frac{1}{4}$

ns(ξ)±cs(ξ)

$\frac{1}{4}(1-m^2)$

$\frac{1}{2}(1+m^2)$

$\frac{1}{4}(1-m^2)$

nc(ξ)±sc(ξ)

$\frac{m^2}{4}$

$\frac{1}{2}(m^2-2)$

$\frac{m^2}{4}$

sn(ξ)±icn(ξ)

$\frac{m^2}{4}$

$\frac{1}{2}(m^2-2)$

$\frac{1}{4}$

ns(ξ)±ds(ξ)

$\frac{m^2}{4}$

$\frac{1}{2}(m^2-2)$

$\frac{m^2}{4}$

$\sqrt{1-m^2}sd(\xi )\pm cd(\xi )$

$\frac{1}{4}$

$\frac{1}{2}(1-m^2)$

$\frac{1}{4}$

$mcd(\xi )\pm i\sqrt{1-m^2}nd(\xi )$

$\frac{1}{4}$

$\frac{1}{2}(1-2m^2)$

$\frac{1}{4}$

msn(ξ)±idn(ξ)

$\frac{1}{4}$

$\frac{1}{2}(1-m^2)$

$\frac{1}{4}$

$\sqrt{1-m^2}sc(\xi )\pm dc(\xi )$

$\frac{1}{4}(m^2-1)$

$\frac{1}{2}(1+m^2)$

$\frac{1}{4}(m^2-1)$

msd(ξ)±nd(ξ)

$\frac{1}{4}$

$\frac{1}{2}(m^2-2)$

$\frac{m^2}{4}$

sn(ξ)/(1 ± dn(ξ))