The Rise of Solitons in Sine-Gordon Field Theory: From Jacobi Amplitude   to Gudermannian Function
  We show how the famous soliton solution of the classical sine-Gordon field theory in $(1+1)$-dimensions may be obtained as a particular case of a solution expressed in terms of the Jacobi amplitude, which is the inverse function of the incomplete elliptic integral of the first kind. 
Leonardo Mondaini
hep-th