Fig. 6

The Jacobi elliptic function \(\textrm{cn}(\zeta ,m)\) in Eq. (19) is illustrated in 2D, showing the evolution from singular periodic to regular periodic waves and ultimately to a bright soliton as the modulus \(m \rightarrow 1\), with \(d_2 = 2\), \(\alpha _3 = -3\), \(\alpha _4 = 2\), and \(\alpha _5 = -1\), at three representative times \(t = 1, 5, 10.\).